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ENGINEERING    MATHEMATICS 


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ENGINEERING  MATHEMATICS 


A  SERIES  OF  LECTURES  DELIVERED 
AT  UNION  COLLEGE 


BY 

CHARLES   PROTEUS  STEINMETZ,  A.M.,  Ph.D. 

/i 

PAST    PRESIDENT 
AMERICAN    INSTITUTE    OF    ELECTRICAL    ENGINEERS 


FIRST   EDITION 

SECOND     IMMIr.-S.OS,    COKREOTED 


... 


McGRAW-HILL    BOOK    COMPANY 

239   WEST    39TH    STREET.   NEW    YORK 

6  Bouverie  Street,  London,  E.C. 

1911 


i  N  1    O  Wf^t 


Copyright,  1011, 


BY 


McGRAW-HILL    BOOK   COMPANY 


PREFACE. 


The  following  work  embodies  the  subject-matter  of  a  lecture 
course  which  I  have  given  to  the  junior  and  senior  electrical 
engineering  students  of  Union  University  for  a  number  of 
years. 

It  is  generally  conceded  that  a  fair  knowledge  of  mathe- 
matics is  necessary  to  the  engineer,  and  especially  the  electrical 
engineer.  For  the  latter,  however,  some  branches  of  mathe- 
matics are  of  fundamental  importance,  as  the  algebra  of  the 
general  number,  the  exponential  and  trigonometric  series,  etc., 
which  are  seldom  adequately  treated,  and  often  not  taught  at 
all  in  the  usual  text-books  of  mathematics,  or  in  the  college 
course  of  analytic  geometry  and  calculus  given  to  the  engineer- 
ing students,  and,  therefore,  electrical  engineers  often  possess 
little  knowledge  of  these  subjects.  As  the  result,  an  electrical 
engineer,  even  if  he  possess  a  fair  knowledge  of  mathematics, 
may  often  find  difficulty  in  dealing  with  problems,  through  lack 
of  familiarity  with  these  branches  of  mathematics,  which  have 
become  of  importance  in  electrical  engineering,  and  may  also 
find  difficulty  in  looking  up  information  on  these  subjects. 

'In  the  same  way  the  college  student,  when  beginning  the 
study  of  electrical  engineering  theory,  after  completing  his 
general  course  of  mathematics,  frequently  finds  himself  sadly 
deficient  in  the  knowledge  of  mathematical  subjects,  of  which 
a  complete  familiarity  is  required  for  effective  understanding 
of  electrical  engineering  theory.  It  was  this  experience  which 
led  me  some  years  ago  to  start  the  course  of  lectures  which 
is  reproduced  in  the  following  pages.  I  have  thus  attempted  to 
bring  together  and  discuss  explicitly,  with  numerous  practical 
applications,  all  those  branches  of  mathematics  which  are  of 
special  importance  to  the  electrical   engineer.     Added  thereto 


257941 


vi  PREFACE. 

are  a  number  of  subjects  which  experience  has  shown  me 
to  be  important  for  the  effective  and  expeditious  execution  of 
electrical  engineering  calculations.  Mere  theoretical  knowledge 
of  mathematics  is  not  sufficient  for  the  engineer,  but  it  must 
be  accompanied  by  ability  to  apply  it  and  derive  results — to 
can-)-  out  numerical  calculations.  It  is  not  sufficient  to  know 
how  a  phenomenon  occurs,  and  how  it  may  be  calculated,  but 
very  often  there  is  a  wide  gap  between  this  knowledge  and  the 
ability  to  carry  out  the  calculation;  indeed,  frequently  an 
attempt  to  apply  the  theoretical  knowledge  to  derive  numerical 
results  leads,  even  in  simple  problems,  to  apparently  hopeless 
complication  and  almost  endless  calculation,  so  that  all  hope 
of  getting  reliable  results  vanishes.  Thus  considerable  space 
has  been  devoted  to  the  discussion  of  methods  of  calculation, 
the  use  of  curves  and  their  evaluation,  and  other  kindred 
subjects  requisite  for  effective  engineering  work. 

Thus  the  following  work  is  not  intended  as  a  complete 
course  in  mathematics,  but  as  supplementary  to  the  general 
college  course  of  mathematics,  or  to  the  general  knowledge  of 
mathematics  which  every  engineer  and  really  every  educated 
man  should  possess. 

In  illustrating  the  mathematical  discussion,  practical 
examples,  usually  taken  from  the  field  of  electrical  engineer- 
ing, have  been  given  and  discussed.  These  are  sufficiently 
numerous  that  any  example  dealing  with  a  phenomenon 
with  which  the  reader  is  not  yet  familiar  may  be  omitted  and 
taken  up  at  a  later  time. 

As  appendix  is  given  a  descriptive  outline  of  the  intro- 
duction to  the  theory  of  functions,  since  the  electrical  engineer 
should  be  familiar  with  the  general  relations  between  the 
different  functions  which  he  meets. 

In  relation  to  "  Theoretical  Elements  of  Electrical  Engineer- 
ing," "Theory  and  Calculation  of  Alternating  Current  Phe- 
nomena," and  "Theory  and  Calculation  of  Transient  Electric 
Phenomena,"  the  following  work  is  intended  as  an  introduction 
and  explanation  of  the  mathematical  side,  and  the  most  efficient 
method  of  study,  appears  to  me,  to  start  with  "  Electrical 
Engineering  Mathematics,"  and  after  entering  its  third 
chapter,  to  take  up  the  reading  of  the  first  section  of  "Theo- 
retical Elements,"  and  then  parallel  (lie  study  of  "Electrical 


PREFACE.  vii 

Engineering  Mathematics,"  "  Theoretical  Elements  of  Electrical 
Engineering,"  and  "  Theory  and  Calculation  of  Alternating 
Current  Phenomena,"  together  with  selected  chapters  from 
"  Theory  and  Calculation  of  Transient  Electric  Phenomena," 
and  after  this,  once  more  systematically  go  through  all  four 
books. 

Charles  P.  Steinmetz. 

Schenectady,  N.  Y., 
December,  1910. 


CONTENTS. 


PAOE 

Preface v 

CHAPTER    I.     THE   GENERAL    NUMBER. 

A.  The  System  of  Numbers. 

1.  Addition  and  Subtraction.     Origin  of  numbers.     Counting  and 

measuring.     Addition.     Subtraction  as   reverse  operation  of 
addition 1 

2.  Limitation  of  subtraction.     Subdivision  of  the  absolute  numbers 

into  positive  and  negative 2 

3.  Negative  number  a  mathematical  conception  like  the  imaginary 

number.     Cases  where  the  negative  number  has  a  physical 
meaning,  and  cases  where  it  has  not 4 

4.  Multiplication  and  Division.     Multiplication  as  multiple  addi- 

tion.    Division  as  its  reverse  operation.     Limitation  of  divi- 
sion         G 

5.  The  fraction  as  mathematical  conception.     Cases  where  it  has  a 

physical  meaning,  and  eases  where  it  has  not 8 

G.  Involution  and  Evolution.  Involution  as  multiple  multiplica- 
tion. Evolution  as  its  reverse  operation.  Negative  expo- 
nents         9 

7.  Multiple  involution  leads  to  no  new  operation 10 

8.  Fractional  exponents 10 

9.  Irrational  Numbers.     Limitation  of  evolution.     Endless  decimal 

fraction.     Rationality  of  the  irrational  number 11 

10.  Quadrature  numbers.     Multiple  values  of  roots.     Square  root  of 

negative  quantity  representing  quadrature  number,  or  rota- 
tion by  90° 13 

11.  Comparison   of   positive,    negative    and    quadrature    numbers. 

Reality  of  quadrature  number.     Cases  where  it  has  a  physical 

meaning,  and  cases  where  it  has  not 11 

11'.   General  Numbers.     Representation  of  the  plane  by  the  general 

number.     Its  relation  to  rectangular  coordinates 16 

13.  Limitat i< in  <  >f  algebra  by  the  general  number.     Root.-  of  the  unit. 

Number  of  such  roots,  and  their  relation 18 

14.  The  two  reverse  operations  of  involution 19 

ix 


COX  TEXTS. 


PAGE 


15.  Logarithmation.    Relation  between  logarithm  and  exponent  of 

involution.  Reduction  toother  base.  Logarithm  of  negative 
quantity ■ 20 

16.  Quaternions.     Vector  calculus  of  space 22 

17.  Space  rotors  and  their  relation.     Super  algebraic  nature  of  space 

analysis 22 

P>.   Algebra  of  the  General  Number  of  Complex  Quantity. 

Rectangular  and  Polar  Coordinates 25 

JS.    Powers  of  /.     Ordinary  or  real,  and  quadrature  or  imaginary 

number.     Relations 25 

L9.  Conception  of  general  number  by  point  of  plane  in  rectangular 
coordinates;  in  polar  coordinates.  Relation  between  rect- 
angular and  polar  form 2(1 

20.  Addition  and  Subtraction.     Algebraic  and  geometrical  addition 

and  subtraction.  Combination  and  resolution  by  parallelo- 
gram law 28 

21.  1  ^notations 30 

22.  Sign  of  vector  angle.     Conjugate  and  associate  numbers.     Vec- 

t  <  >r  analysis 30 

23.  Instance  of  steam  path  of  turbine 33 

24.  Multiplication.      Multiplication   in   rectangular  coordinates....      38 
"Jo.    Multiplication  in  polar  coordinates.     Vector  and  operator 38 

26.  Physical  meaning  of  result  of  algebraic  operation.     Representa- 

tion of  result 40 

27.  Limit. it  ion  of  application  of  algebraic  operations  to  physical 

quantities,  and  of  the  graphical  representation  of  the  result. 
Graphical  representation  of  algebraic  operations  between 
current,  voltage  and  impedance 40 

28.  Representation  of  vectors  and  of  operators 42 

2'.).    Division.      Division  in  rectangular  coordinates 42 

30.  Division  in  polar  coordinates 43 

31.  Involution  and  Evolution.      Use  of  polar  coordinates 44 

32.  Multiple  values  of  the  result  of  evolution.    Their  location  in  the 

plane  of  the  general  number.      Polyphase  and  n  phase  systems 

of  numbers • 45 

:;:',.    The  //  values  of  V]  and  their  relation 46 

:;!.   Evolution  in  rectangular  coordinates.     Complexity  of  resull  ...      17 

35.  Reducti f  products  and  tract  ions  of  general  numbers  by  polar 

representation.     Instance is 

36.  Exponential  representations  of  general  numbers.    The  different 

forms  of  i  he  general  number 1{.) 

:',~.   Instance  ol   use  of  exponential  form  in  solution  of  differential 

equation 50 


CONTENTS.  xi 

PAliE 

38.  Logarithmation.     Resolution    of   the    logarithm   of   a   general 

number 51 

CHAPTER   II.     THE    POTENTIAL    SERIES    AND  EXPONENTIAL 

FUNCTION. 

A.  General. 

39.  The  infinite  series  of  powers  of  x 52 

40.  Approximation  by  series .">•'! 

41.  Alternate  and  one-sided  approximation 5  I 

42.  Convergent  and  divergent  series .V> 

43.  Range  of  convergency.     Several   series  of  different   ranges  for 

same  expression   56 

44  Discussion  of  convergency  in  engineering  applications 57 

45.  Use  of  series  for  approximation  of  small  terms.     Instance  of 

electric  circuit 58 

46.  Binomial  theorem  for  development  in  series.      Instance  of  in- 

ductive circuit 59 

47.  Necessity  of  development  in  series.     Instance  of  arc  of  hyperbola  60 

48.  Instance  of  numerical  calculation  of  log  (1  +  x) 63 

B.  Differential  Equations. 

49.  Character  of  most  differential  equations  of  electrical  engineering. 

Their  typical   forms 64 

dy 

50.  —  =  y.     Solution   by  scries,  by  method   ol  indeterminate    co- 
ax 

efficients 65 

dz 

51.  —  =  az.     Solution  by  indeterminate  coefficients 68 

ax 

52.  Integration  constant  and  terminal  conditions lis 

53.  Involution  of  solution.     Exponential  function 70 

54.  Instance  of  rise  of  field  current  in  direct  current  shunt  motor  .  .  72 

55.  Evaluation  of  inductance,  and  numerical  calculation 75 

56.  Instance  of  condenser  discharge  through  resistance 76 

57.  Solution  of  —  =  ay  by  indeterminate  coefficients,  by  exponential 

function 78 

58.  Solution  by  trigonometric  functions SI 

59.  Relations  between  trigonometric  functions  and  exponential  func- 

tions with  imaginary  exponent,  and  inversely 83 

60.  Instance  of  condenser  discharge  through  inductance.     The  two 

integration  constants  and  terminal  conditions 84 

61.  Effect  of  resistance  on  the  discharge.     The  general  differential 

equation 86 


xii  (OX  TEXTS. 

PAGE 

62.  Solution  of  the  general  differentia]  equation  by  means  of  the 

exponential    function,     by    the    method    of    indeterminate 
ci  (efficients -     86 

63.  [nstance  of  condenser  discharge  through  resistance  and  induc- 

tance.    Exponential  solution  and  evaluation  of  constants.  .  ..     S8 

64.  Imaginary  exponents  of  exponential  functions.     Reduction  to 

trigonometric  functions.     The  oscillating  functions 91 

65.  Explanation  of  tables  of  exponential   functions) 92 

CHAPTER    III.     TRIGONOMETRIC    SERIES 

A.  Trigonometric  Functions. 

66.  Definition  of  trigonometric  functions  on  circle  and  right  triangle  94 

07.  Sign  of  functions  in  different  quadrants '.).-> 

lis.   Relations  between  sin.  cos,  tan  and  cot 97 

69.  Negative,  supplementary  and  complementary  angles 98 

70.  Angles  (x±n)  and  (z±^) 100 

71.  Relations  between  two  angles,  and  between  angle  and  double 

.'ingle 102 

72.  Differentiation    and    integration    of    trigonometric    functions. 

Definite  integrals 103 

73.  The  binomial  relations 104 

71.    Polyphase  relations 104 

7.">.  Trigonometric  formulas  of  the  triangle IO.j 

1>.   Trigonometric  Series. 

T' '•-   Constant,  transient  and  periodic  phenomena.     Univalent    peri- 
odic   function    represented   by   trigonometric  series 100 

77.  Alternating  sine  waves  and  distorted  waves 107 

78.  Evaluation  of  the  Constants  from  Instantaneous  Values.     Cal- 

culation of  constant  term  of  series IDS 

7!).    ( !alculation  of  cos-coefficients 110 

80.   Calculation  of  sin-coefficients 113 

M  .    Instance  of  calculating  11th  harmonic  of  generator  wave Ill 

82.    Discussion.     Instance  of  complete  calculation  of  pulsating  cur- 
rent wave IIC, 

V..    Alternating  waves  as  symmetrical  waves.     Calculation  of  sym- 
metrical wave    117 

8  1.   Separation  of  odd  and  even  harmonics  and  of  constant  term  .  ..  120 

85.  Separation  of  sine  and  cosine  components 121 

86.  Separation  of  wave  into  constant  term  and    1  component   waves  122 

87.  Discussion  of  calculation 123 

88.  Mechanism  of  calculal  ion p_>4 


CONTENTS.  xiii 

PAGE 

S9.   Instance  of  resolution  of  the  annual  temperature  curve 125 

!)().    Constants  and  equation  of  temperature  wave 131 

91.  Discussion  of  temperature  wave 132 

C.  Reduction  of  Trigonometric  Series  by  Polyphase  Relation. 

92.  Method  of  separating  certain    classes  of   harmonies,    and   its 

limitation 1 34 

93.  Instance  of  separating  the  3d  and  9th  harmonic  of  transformer 

exciting  current 130 

1).  Calculation  of  Trigonometric  Series  from  other  Trigono- 
metric Series. 

94.  Instance  of  calculating  current  in  long  distance  transmission  line, 

due  to  distorted  voltage  wave  of  generator.     Line  constants.  .    139 

95.  Circuit  equations,  and  calculation  of  equation  of  current 141 

90.   Effective  value  of  current,  and  comparison  with  the  current 

produced  by  sine  wave  of  voltage 143 

97.  Voltage  wave  of  reactance  in  circuit  of  this  distorted  current  ...    14") 

CHAPTER   IV.     MAXIMA   AND   MINIMA 

98.  Maxima  and  minima  by  curve  plotting.     Instance  of  magnetic 

permeability.     Maximum  power  factor  of  induction  motor  as 
function  of  load 147 

99.  Interpolation  of  maximum  value  in  method  of  curve  plotting. 

Error  in  case  of  unsymmetrical  curve.     Instance  of  efficiency 

of  steam  turbine  nozzle.     Discussion 149 

100.  Mathematical    method.     Maximum,    minimum    and    inflexion 

point.     Discussion 152 

101.  Instance:     Speed    of    impulse    turbine    wheel    for    maximum 

efficiency.     Current  in  transformer  for  maximum  efficiency.  154 

102.  Effect  of  intermediate  variables.     Instance:    Maximum  power 

in  resistance  shunting  a  constant  resistance  in  a  constant  cur- 
rent circuit 155 

103.  Simplification  of  calculation  by  suppression  of  unnecessary  terms, 

etc.     Instance 157 

104.  Instance:   Maximum  non-inductive  load  on  inductive  transmis- 

sion line.     Maximum  current  in  line 158 

105.  Discussion   of   physical   meaning  of  mathematical   extremum. 

Instance 160 

106.  Instance:   External  reactance  giving  maximum  output  of  alter- 

nator at  constant  external  resistance  and  constant  excitation. 
Discussion 161 

107.  Maximum  efficiency  of  alternator  on  non-inductive  load.     Dis- 

cussion of  physical  limitations 162 


xiv  < 'OX TEXTS. 

108.  Fxtrema  with  several  independenl  variables.     Method  of  mm li 

ematical  calculation,  and  geometrical  meaning 163 

109.  Resistance  and  reactance  of  load  to  give  maximum  output  of 

transmission  line,  at  constant  supply  voltage 165 

110.  Discussion  of  physical  limitations 107 

111.  Determination  of  extrema  by  plotting  curve  of  differential  quo- 

tient.    Instance:    Maxima  of  current  wave  of  alternator  of 
distorted  voltage  on  transmission  line 168 

112.  Graphical  calculation  of  differential  curve  of  empirical  curve, 

for  determining  extrema 170 

113.  Instance:  Maximum  permeability  calculation 170 

114.  Grouping  of  battery  cells  for  maximum  power  in  constant  resist- 

ance      171 

115.  Voltage  of  transformer  to  give   maximum  output  at  constant 

loss 173 

116.  Voltage  of  transformer,  at  constant  output,  to  give  maximum 

efficiency  at  full  load,  at  half  load 1 74 

117.  Maximum    value    of    charging    current    of    condenser    through 

inductive  circuit  (a)  at  low  resistance;   (b)  at  high  resistance.  175 

1 1 8.  At  what  output  is  the  efficiency  of  an  induction  generator  a  max- 

imum?      177 

1 19.  Discussion  of  physical  limitations.     Maximum  efficiency  at  con- 

stant current  output 17S 

120.  Method  of  Least  Squares.     Relation  of  number  of  observa- 

tions   to    number    of    constants.      Discussion    of    errors    of 

<  ibservation 1 70 

121.  Probability   calculus   and  the  minimum  sum  of  squares  of  the 

err<  >rs : P<  I 

122.  The  differential  equations  of  the  sum  of  least  squares 182 

123.  Instance:     Reduction   of   curve   of   power  of   induction  motor 

running    light,    into    the    component     losses.      Discussion    of 
results 1S2 


CHAPTER    V.     METHODS   OF   APPROXIMATION 

124.  Frequency  of  small  quantities  in  electrical  engineering  problems. 

Instances.     Approximation  by  dropping  terms  of  higher  order.  187 

125.  Multiplication  of  terms  with  small  quantities 188 

126.  Instance  of  calculation  of  power  of  direct  current  shunt  motor  .  189 

127.  Small  quantities  in  denominator  of  fractions 100 

128.  Instance  of  calculation  of  induction  motor  current,  as  function 

Of  slip 191 


CONTENTS.  xv 


PAGE 


12!).  Use  of  binomial  series  in  approximations  of  powers  and  roots, 

and  in  numerical  calculations 193 

130.  Instance  of  calculation  of  current  in  alternating  circuit  of  low- 

inductance.     Instance  of  calculation  of  short  circuit  current 

of  alternator,  as  function  of  speed 195 

131.  Use  of  exponential  series  and  logarithmic  series  in  approxima- 

tions      190 

132.  Approximations  of  trigonometric  functions 198 

133.  McLaurin's  and  Taylor's  series  in  approximations 198 

134.  Tabulation  of  various  infinite  series  and  of  the  approximations 

derived  from  them 199 

135.  Estimation    of   accuracy   of    approximation.     Application    to 

short  circuit  current  of  alternator 200 

136.  Expressions  which  are  approximated  by  (1  +s)  and  by  (1  — s).  .   201 

137.  Mathematical  instance  of  approximation 203 

138.  Equations  of  the  transmission*  Line.      Integration  of  the 

differential  equations 204 

139.  Substitution  of  the  terminal  conditions 205 

140.  The  approximate  equations  of  the  transmission  line 200 

141.  Numerical    instance.     Discussion    of    accuracy    of    approxima- 

tion       207 

CHAPTER    VI.     EMPIRICAL    CURVES 
A.  (  Ikveral. 

142.  Relation    between   empirical  curves,  empirical   equations  and 

rational  equations 209 

143.  Physical  nature  of  phenomenon.     Points  at  zero  and  at  infinity. 

Periodic  or  non-periodic.     Constant  term.--.     Change  of  curve 

law.     Scale 210 

P>.  Non-Periodic   Curves. 

144.  Potential  Series.     Instance  of  core-loss  curve 212 

145.  Rational  and  irrational  use  of  potential  series.     Instance  of  fan 

motor  torque.     Limitations  of  potential  series 214 

140.  Parabolic  and  Hyperbolic  Curves.  Various  shapes  of  para- 
bolas and  of  hyperbolas 210 

147.  The  characteristic  of  parabolic  and  hyperbolic  curves.     Its  use 

and  limitation  by  constant  terms 223 

148.  The  logarithmic  characteristic.     Its  use  and  limitation 224 

14!).  Exponential   and   Looarithmic   Curves.      The  exponential 

function 227 

150.  Characteristics  of  the  exponential  curve,  their  use  and  limitation 
by  constant  term.  Comparison  of  exponential  curve  and 
hyperbola 22S 


xvi  CONTENTS. 

PAGE 

151.  Double  exponential  functions.     Various  shapes  thereof 231 

152.  Evaluation    of    Empirical    Curves.    General    principles   of 

investigation  of  empirical  curves 233 

153.  Instance:   The  volt-ampere  characteristic  of  the  tungsten  lamp, 

reduced   to   parabola    with   exponent  0.0.     Rationalized  by 
reduction  to  radiation  law 235 

154.  'The  volt-ampere  characteristic  of  the  magnetite  arc,  reduced 

to  hyperbola  with  exponent  0.5 23S 

155.  Change  of  electric   current    with   change  of  circuit   conditions, 

reduced  to  double  exponential  function  of  time 241 

156.  Rational   reduction  of    core-loss  curve  of  paragraph   144,  by 

parabola  with  exponent  1.6 244 

157.  Reduction  of  magnetic  characteristic,  for  higher  densities,  to 

hyperbolic  curve.     Instance  of  the   investigation  of  a  hys- 
teresis curve  of  silicon  steel 246 

C.  Periodic  Curves. 

158.  Distortion  of  sine  wave  by  lower  harmonics 255 

I. V.).  Ripples  and  nodes  caused  by  higher  harmonics.     Incommen- 
surable waves 255 


CHAPTER    VII.     NUMERICAL    CALCULATIONS 

Mil).  Method  of  Calculation.     Tabular  form  of  calculation 2">S 

1(11 .  Instance  of  transmission  line  regulation 230 

l(i'_\  Exactness  of  Calculation.     Degrees  of  exactness:    magni- 
tude, approximate,  exact l'Ol' 

103.  Number  of  decimals 204 

164.  Intelligibility  of  Enc.ineerin<;   Data.     Curve  plotting  for 

showing  shape  of  function,  and  for  record  of  numerical  values  200 
10.").  Scale  of  curves.      Principles 269 

166.  Completeness  of  record 270 

167.  Reliability    of    Numerical   Calculations.       Necessity   of 

reliability  in  engineering  calculations 271 

168.  Methods  of  checking  calculations.      Curve  plotting 272 

273 


169.  Some  frequent  errors. 


APPENDIX    A.     NOTES    ON    THE    THEORY    OF    FUNCTIONS 

A.  ( rENEB  \i.  Functions. 

170.  [mplicil     analytic     function.        Explicit      analytic     function. 

Reverse  function 27") 

171.  Rational     function.     Integer     function.     Approximations    by 

Tavlor's  Theorem 270 


CONTENTS.  xvil 

PA<SK 

172.  Abelian  integrals  and  Abclian  functions.     Logarithmic  integral 

and  exponential  function 277 

173.  Trigonometric  integrals  and  trigonometric  functions.     Hyper- 

bolic integrals  and  hyperbolic  functions 278 

174.  Elliptic  integrals  and  elliptic  functions.  Their  double  periodicity  279 

175.  Theta  functions.     Hyperelliptic  integrals  and  functions 281 

176.  Elliptic  functions  in  the  motion  of  the  pendulum  and  the  surging 

of  synchronous  machines 282 

177.  Instance  of  the  arc  of  an  ellipsis 282 

13.    Special  Functions. 

178.  Infinite  summation  series.      Infinite  product   series 283 

179.  Functions  by  integration.     Instance  of  the  propagation  func- 

tions of  electric  waves  and  impulses 284 

180.  Functions  defined  by  definite  integral? 286 

181.  Instance  of  the  gamma  function 287 

C.  Exponential,  Trigonometric  and  Hyperbolic  Functions. 

1S2.  Functions  of  real  variables 287 

183.  Functions  of  imaginary  variables 289 

1S4.>  Functions  of  complex  variables 289 

185.  Relations 290 


APPENDIX   B.     TABLES 

Table  I.     Three  decimal  exponential  functions 293 

Table  II.     Logarithms  of  exponential  functions 294 

Exponential  function 295 

Hyperbolic  f unct  ions 295 


ENGINEERING  MATHEMATICS. 


CHAPTER   I. 
THE  GENERAL  NUMBER. 

A.  THE  SYSTEM  OF  NUMBERS. 
Addition  and  Subtraction. 

i.  From  the  operation  of  counting  and  measuring  arose  the 
art  of  figuring,  arithmetic,  algebra,  and  finally,  more  or  less, 
the  entire  structure  of  mathematics. 

During  the  development  of  the  human  race  throughout  the 
ages,  which  is  repeated  by  every  child  during  the  first  years 
of  life,  the  first  conceptions  of  numerical  values  were  vague 
and  crude:  many  and  few,  big  and  little,  large  and  small. 
Later  the  ability  to  count,  that  is,  the  knowledge  of  numbers, 
developed,  and  last  of  all  the  ability  of  measuring,  and  even 
up  to-day,  measuring  is  to  a  considerable  extent  done  by  count- 
ing: steps,  knots,  etc. 

From  counting  arose  the  simplest  arithmetical  operation — 
addition.    Thus  we  may  count  a  bunch  of  horses: 

1,  2,  3,  4,  5, 

and  then  count  a  second  bunch  of  horses, 

1    9    3- 

now  put  the  second  bunch  together  with  the  first  one,  into  one 
bunch,  and  count  them.     That  is,  after  counting  the  horses 


■ 

2  ENGINEERING  MATHEMATICS. 

of  the  first   bunch,  we  continue  to  counl   those  of  the  second 
hunch,  thus : 

1,  2,  3,  4,  5,-6,  7,  8; 

which  gives  addition, 

5+3=8; 

or,  in  general, 

a  +  b  =  r. 

We  may  take  away  again  the  second  bunch  of  horses,  that 
means,  we  count  the  entire  bunch  of  horses,  and  then  count 
off  those  we  take  away  thus: 

.     1,2,  3,  4,  5,  6,  7,  8-7,  6,  5; 

which  gives  subtraction, 

8-3-5; 
or,  in  general, 

x—b  =  a. 

The  reverse  of  putting  a  group  of  things  together  with 
another  group  is  to  take  a  group  away,  therefore  subtraction 
is  the  reverse  of  addition. 

2.  Immediately  we  notice  an  essential  difference  between 
addition  and  subtraction,  which  may  be  illustrated  by  the 
following  examples: 

Addition:  5  horses +3  horses  gives  8  horses, 
Subtraction:  5  horses— 3  horses  gives  2  horses, 
Addition:  5  horses  -f -7  horses  gives  12  horses, 
Subtraction:  5  horses— 7  horses  is  impossible. 

From  the  above  it  follows  that  we  can  always  add,  but  we 
cannot  always  subtract;  subtraction  is  not  always  possible; 
it  is  not,  when  the  number  of  things  which  we  desire  to  sub- 
tract is  greater  than  the  number  of  things  from  which  we 
desire  to  subtract. 

The  same  relation  obtains  in  measuring;  we  may  measure 
a  distance  from  a  starting  point  A  ( Fig.  L),  for  instance  in  steps, 
and  then  measure  a  second  distance,  and  get  the  total  distance 
from  the  starting  noint   bv  addition:    5  steps,  from  A  to  B. 


THE  GENERAL  X UMBER.  3 

then  3  stops,  from  B  to  C,  gives  the  distance  from  /I  to  C,  as 
8  steps. 

5  steps +3  steps  =8  steps; 


12        3-15678 

(fc 1 1 1 1 CD  I I  (D 

A  B  C 

Fig.  1.     Addition. 

or,  we  may  step  off  a  distance,  and   then  step  back,  that  is, 
subtract  another  distance,  for  instance  (Fig.  2), 

5  steps  —3  steps  =  2  steps: 

that  is,  going  5  steps,  from  A  to  B,  and  then  3  steps  back, 
from  B  to  C,  brings  us  to  C,  2  steps  away  from  A. 


1        2        3        4        5 
<t)         I CD         l 1 CD 

AC  B 

Fig.  2.     Subtraction. 

Trying  the  case  of  subtraction  which  was  impossible,  in  the 
example  with  the  horses,  5  steps— 7  steps  =  ?  We  go  from  the 
starting  point,  A,  5  steps,  to  B,  and  then  step  back  7  steps; 
here  we  find  that  sometimes  we  can  do  it,  sometimes  we  cannot 
do  it;  if  back  of  the  starting  point  A  is  a  stone  wall,  we  cannot 
step  back  7  steps.  If  A  is  a  chalk  mark  in  the  road,  we  may 
step  back  beyond  it,  and  come  to  C  in  Fig.  3.     In  the  latter  case, 


2        10        12        3        4        5 
-4 1 (p        I 1 1 1         0 


C  A  B 

Fig.  3.     Subtraction.  Negative  Result. 

at  C  we  are  again  2  steps  distant  from  the  starting  point,  just 
as  in  Fig.  2.     That  is, 

5-3  =  2     (Fig.  2), 

5-7  =  2     (Fig.  3). 

In  the  case  where  we  can  subtract  7  from  5,  we  get  the  same 
distance  from  the  starting  point  as  when  we  subtract  3  from  5, 


4  ENGINEERING  MATHEMATICS. 

but  the  distance  AC  in  Fig.  3,  while  the  same,  2  steps,  as 
in  Fig.  2,  is  different  in  character,  the  one  is  toward  the  left, 
the  other  toward  the  right.  That  means,  we  have  two  kinds 
of  distance  units,  those  to  the  right  and  those  to  the  left,  and 
have  to  find  some  way  to  distinguish  them.  The  distance  2 
in  Fig.  3  is  toward  the  left  of  the  starting  point  A,  that  is, 
in  that  direction,  in  which  we  step  when  subtracting,  and 
it  thus  appears  natural  to  distinguish  it  from  the  distance 
2  in  Fig.  2,  by  calling  the  former— 2,  while  we  call  the  distance 
AC  in  Fig.  2:  +2,  since  it  is  in  the  direction  from  A,  in  which 
we  step  in  adding. 

This  leads  to  a  subdivision  of  the  system  of  absolute  numbers, 

1    ">    3 

into  two  classes,  positive  numbers, 

+  1,    +2,    +3,  .  ..; 
and  negative  numbers, 

-1,    -2,    -3,  ...; 

and  by  the  introduction  of  negative  numbers,  we  can  always 
carry  out  the  mathematical  operation  of  subtraction: 

c  —  b  =  a, 

and,  if  b  is  greater  than  c,  a  merely  becomes  a  negative  number. 

3.  We  must  therefore  realize  that  the  negative  number  and 
the  negative  unit,  -1,  is  a  mathematical  fiction,  and  not  in 
universal  agreement  with  experience,  as  the;  absolute  number 
found  in  the  operation  of  counting,  and  the  negative  number 
does  not  always  represent  an  existing  condition  in  practical 
experience. 

In  the  application  of  numbers  to  the  phenomena  of  nature, 
we  sometimes  find  conditions  where  we  can  give  the  negative 
number  a  physical  meaning,  expressing  a  relation  as  the 
reverse  to  the  positive  number;  in  other  cases  we  cannot  do 
this.  For  instance,  5  horses -7  horses=  -2  horses  has  no 
physical  meaning.  There  exist  no  negative  horses,  and  at  the 
best  we  could  only  express  the  relation  by  saying,  5  horses— 7 
horses  is  impossible,  2  horses  are  missing. 


THE  GENERAL  NUMBER.  5 

In  the  same  way,  an  illumination  of  5  foot-candles,  lowered 
by  3  foot-candles,  gives  an  illumination  of  2  foot-candles,  thus, 

5  foot-candles —3  foot-candles  =  2  foot-candles. 

If  it  is  tried  to  lower  the  illumination  of  5  foot-candles  by  7 
foot-candles,  it  will  be  found  impossible;  there  cannot  be  a 
negative  illumination  of  2  foot-candles;  the  limit  is  zero  illumina- 
tion, or  darkness. 

From  a  string  of  5  feet  length,  we  can  cut  off  3  feet,  leaving 

2  feet,  but  we  cannot  cut  off  7  feet,  leaving  —2  feet  of  string. 

In  these  instances,  the  negative  number  is  meaningless 
a  mere  imaginary  mathematical  fiction. 

If  the  temperature  is  5  deg.  cent,  above  freezing,  anil  falls 

3  deg.,  it  will  be  2  deg.  cent,  above  freezing.  If  it  falls  7  deg. 
it  will  be  2  deg.  cent,  below  freezing.  The  one  case  is  just  as 
real  physically,  as  the  other,  and  in  this  instance  we  may 
express  the  relation  thus: 

+5  deg.  cent.  —3  deg.  cent.=  +2  deg.  cent., 

+  5  deg.  cent.    -7  deg.  cent.  =  —2  deg.  cent.; 

that  is,  in  temperature  measurements  by  the  conventional 
temperature  scale,  the  negative  numbers  have  just  as  much 
physical  existence  as  the  positive  numbers. 

The  same  is  the  case  with  time,  we  may  represent  future 
time,  from  the  present  as  starting  point,  by  positive  numbers, 
and  past  time  then  will  be  represented  by  negative  numbers. 
But  we  may  equally  well  represent  past  time  by  positive  num- 
bers, and  future  times  then  appear  as  negative  numbers.  In 
this,  and  most  other  physical  applications,  the  negative  number 
thus  appears  equivalent  with  the  positive  number,  anil  inter- 
changeable: we  may  choose  any  direction  as  positive,  and 
the  reverse  direction  then  is  negative.  Mathematically,  how- 
ever, a  difference  exists  between  the  positive  and  the  negative 
number;  the  positive  unit,  multiplied  by  itself,  remains  a  pos- 
itive unit,  but  the  negative  unit,  multiplied  with  itself,  does 
not  remain  a  negative  unit,  but  becomes  positive: 

(  +  l)x(  +  l)  =  (  +  l); 

(-1)X(-1)  =  (  +  1),  and  not  =(-1). 


6  ENGINEERING  MA  THEM .  1  Til 'S. 

Starting  from  5  dog.  northern  latitude  and  going  7  deg. 
south,  brings  us  to  2  deg.  southern  latitude,  which  may  be 
expresses  thus, 

+  5  deg.  latitude —7  deg.  latitude  =    —2  deg.  latitude. 

Therefore,  in  all  cases,  where  there  arc  two  opposite  direc- 
tions, right  and  left  on  a  line,  north  and  south  latitude,  east 
and  west  longitude,  future  and  past,  assets  and  liabilities,  etc, 
there  may  be  application  of  the  negative  number;  in  other  cases, 
where  there  is  only  one  kind  or  direction,  counting  horses, 
measuring  illumination,  etc.,  there  is  no  physical  meaning 
which  would  be  represented  by  a  negative  number.  There 
are  still  other  cases,  where  a  meaning  may  sometimes  be  found 
and  sometimes  not;  for  instance,  if  we  have  5  dollars  in  our 
pocket,  we  cannot  take  away  7  dollars;  if  we  have  5  dollars 
in  the  bank,  we  may  be  able  to  draw  out  7  dollars,  or  we  may 
not,  depending  on  our  credit.  In  the  first  case,  5  dollars  -7 
dollars  is  an  impossibility,  while  the  second  case  5  dollars  —7 
dollars  =  2  dollars  overdraft. 

In  any  case,  however,  we  must  realize  that  the  negative 
number  is  not  a  physical,  but  a  mathematical  conception, 
which  may  find  a  physical  representation,  or  may  not,  depend- 
ing on  the  physical  conditions  to  which  it  is  applied.  The 
negative  number  thus  is  just  as  imaginary,  and  just  as  real, 
depending  on  the  case  to  which  it  is  applied,  as  the  imaginary 

number  "v—1,  and  the  only  difference  is,  that  we  have  become 
familiar  with  the  negative  number  at  an  earlier  age,  when1  we 
wer?  less  critical,  and  thus  have  taken  it  for  granted,  become 
familiar  with  it  by  use,  and  usually  do  not  realize  that  it  is 
a  mathematical  conception,  and  not  a  physical  reality.  When 
we  first  learned  it,  however,  it  was  quite  a  step  to  become 
accustomed  to  saying,  5— 7  =—2,  and  not  simply,  5—7  is 
impossible. 

Multiplication  and  Division. 

4.  If  we  have  a  bunch  of  1  horses,  and  another  bunch  of  4 
horses,  and  still  another  bunch  of  -1  horses,  and  add  together 
the  three  bunches  of  4  horses  each,  we  get, 

I  horses  I   1  horses  1-1  horses   =  12  horses: 


THE  GENERAL  NUMBER.  7 

or,  as  we  express  it, 

3x4  horses  =  12  horses. 

The  operation  of  multiple  addition  thus  leads  to  the  next 
operation,  multiplication.  Multiplication  is  multiple  addi- 
tion, 

&  X  a = c, 

thus  means 

a  \-a+a  +  .  .  .  {!>  terms)     c. 

Just  like  addition,  multiplication  can  always  be  carried 
out. 

Three  groups  of  4  horses  each,  give  12  horses.  Inversely,  if 
we  have  12  horses,  and  divide  them  into  3  equal  groups,  each 
group  contains  4  horses.  This  gives  us  the  reverse  operation 
of  multiplication,  or  division,  which  is  written,  thus: 

12  horses     ,  , 

~ =  4  horses; 

or,  in  general, 

c 

T  =  a. 

o 

If  we  have  a  hunch  of  12  horses,  and  divide  it  into  two  equal 
groups,  we  get  6  horses  in  each  group,  thus: 

12  horses 

^ =  o  horses, 


if  we  divide  unto  4  equal  groups, 

12  horses 


3  horses. 


If  now  we  attempt  to  divide  the  bunch  of  12  horses  into  5  equal 
groups,  we  find  we  cannot  do  it;  if  we  have  2  horses  in  each 
group,  2  horses  are  left  over;  if  we  put  3  horses  in  each  group, 
we  do  not  have  enough  to  make  5  groups;  that  is,  12  horses 
divided  by  5  is  impossible;  or,  as  we  usually  say;  12  horses 
divided  by  5  gives  2  horses  and  2  horses  left  over,  which  is 
written, 

12 

-=-  =  2,  remainder  2. 
5 


8  ENGINEERING  MATHEMA  Tit 'S. 

Thus  it  is  seen  that  the  reverse  operation  of  multiplication, 
or  division,  cannot  always  he  carried  out. 

5.  If  we  have  10  apples,  and  divide  them  into  3,  we  get  3 
apples  in  each  group,  and  one  apple  left  over. 

-q~  =  3,  remainder  1, 

we  may  now  cut  the  left-over  apple  into  3  equal  parts,  in  which 
ease, 

10         1_   t 

In  the  same  manner,  if  we  have  12  apples,  we  can  divide 
into  5,  by  cutting  2  apples  each  into  5  equal  pieces,  and  get 
in  each  of  the  5  groups,  2  apples  and  2  pieces. 

12  1 

—  =  9-1-9  v-  =  23 

To  be  able  to  carry  the  operation  of  division  through  for 
all  numerical  values,  makes  it  necessary  to  introduce  a  new 
unit,  smaller  than  the  original  unit,  and  derived  as  a  part  of  it. 

Thus,  if  we  divide  a  string  of  10  feet  length  into  3  equal 
parts,  each  part  contains  3  feet,  and  1  foot  is  left  over.  One 
foot  is  made  up  of  12  inches,  and  12  inches  divided  into  3  gives 
4  inches;    hence,  10  feet  divided  by  3  gives  3  feet   4  inches. 

Division  leads  us  to  a  new  form  of  numbers:  the  fraction. 

The  fraction,  however,  is  just  as  much  a  mathematical  con- 
ception, which  sometimes  may  be  applicable,  and  sometimes 
not,  as  the  negative  number.  In  the  above  instance  of  12 
horses,  divided  into  5  groups,  it  is  not  applicable. 

12  horses 

^ =  2f  horses 

is  impossible;  we  cannot  have  fractions  of  horses,  and  what 
we  would  get  in  this  attempt  would  be  5  groups,  each  com- 
prising 2  horses  and  some  pieces  of  carcass. 

Thus,  the  mathematical  conception  of  the  fraction  is  ap- 
plicable to  those  physical  quantities  which  can  be  divided  into 
smaller  units,  but  is  not  applicable  to  those,  which  are  indi- 
visible, or  individuals,  as  we  usually  call  them. 


THE  GENERAL  NUMBER.  9 

Involution  and  Evolution. 

6.  If  \vc  have  a  product  of  several  equal  factors,  as, 

4X4X4  =  64, 
it  is  written  as,  43  =  64 ; 

or,  in  general,  ab  =  c. 

The  operation  of  multiple  multiplication  of  equal  factors 
thus  leads  to  the  next  algebraic  operation — involution;  just  as 
the  operation  of  multiple  addition  of  equal  terms  leads  to  the 
operation  of  multiplication. 

The  operation  of  involution,  defined  as  multiple  multiplica- 
tion, requires  the  exponent  b  to  be  an  integer  number;  b  is  the 
number  of  factors. 

Thus  4~3  has  no  immediate  meaning;  it  would  by  definition 
be  4  multiplied  ( —3)  times  with  itself. 

Dividing  continuously  by  4,  we  get,  46^4=45;  45^4=44; 
44-^4  =  43;  etc.,  and  if  this  .successive  division  by  4  is  carried 
still  further,  we  get  the  following  series: 


43     4x4x4 
4"         4 

=  4X4 

=  42 

42     4X4 
4         4 

=  4 

=  4* 

41     4 
T~4 

=  1 

=  4° 

4°     1 

1 

=  4-1 

4  =4 

~4 

A 

4     4 

1 
"4X4 

42 

4"V  ^4 

4       42 

1 

=  4_3=i. 

43' 

4X4X4 

t     1 

or,  in  general, 

a         ab' 

oP  =  l. 


10  ENGINEERING  M .  1  THEM .  \  TJ(  'S. 

Thus,  powers  with  negative  exponents,  as  a~b,  are  the 
reciprocals  of  the  same  powers  with  positive  exponents:  ~jj. 

7.  From  the  definition  of  involution  then  follows, 

abXan  =  ab+n, 

because  ab  means  the  product  of  h  equal  factors  a,  and  an  the 
product  of  n  equal  factors  a,  and  abXan  thus  is  a  product  hav- 
ing b  +  n  equal  factors  a.     For  instance, 

43  X42  =  (4x4x4)  X  (4X4)  =  4-\ 

The  question  now  arises,  whether  by  multiple  involution 
we  can  reach  any  further  mathematical  operation.     For  instance, 

(43)2  =  ?,  . 

may  be  written, 

(43)2  =  43  X  43 

=  (4X4X4)  X  (4X4X4); 

=  46. 

and  in  the  same  manner, 

(ab)n  =  abn; 

that  is,  a  power  ub  is  raised  to  the  nth  power,  by  multiplying 
its  exponent.     Thus  also, 

(ab)n  =  {an)b; 

that  is,  the  order  of  involution  is  immaterial. 

Therefore,  multiple  involution  leads  to  no  further  algebraic 
operations. 

8.  4:i  =  ()4; 

that  is.  the  product  of  3  equal  factors  4,  gives  <>4. 

Inversely,  the  problem  may  be,  to  resolve  (>4  into  a  product 
of  3  equal  factors.  Each  of  the  factors  then  will  be  4.  This 
reverse  operation  of  involution  is  called  evolution,  and  is  written 

thus, 

\V(i4  =  4; 

or,  more  general, 

b/~ 


THE  GENERAL  NUMBER.  11 

Vc  thus  is  defined  as  thai  number  a,  which,  raised  to  the  power 
b,  gives  c;    or,  in  other  words, 

(^)"=c. 

Involution  thus   far  was  defined  only  for  integer  positive 
and  negative  exponents,  and  the  question  arises,  whether  powers 

J_  n 

with  fractional  exponents,  as  ct>    or  c*,  have    any    meaning. 
Writing, 

\Ch)    =c     b  =c1=c, 

it  is  seen  that  c^  is  that  number,  which  raised  to  the  power  h, 

1  .      b  — 
gives  c;    that  is,  c&  is  \  c,  and  the  operation  of  evolution  thus 

can  be  expressed  as  involution  with  fractional   exponent, 

T        b~ 
cb  =  \  c, 

and 


or, 

n 
Cb   =  (rn)  6    ==  \    gtt 


4-  6  — 


and 

Obviously  then, 


(•^7)n=  \  <■". 


6  /  j.         P  /  n 

X/C  =  Cb,        \'C  =  C     , 


1  1  1 


b/c  =  c 


1         b  /-• 
C6 

Irrational  Numbers. 

9.  Involution  with  integer  exponents,  as  43  =  (\4,  can  always 
be  carried  out.  In  many  cases,  evolution  can  also  be  carried 
out.     For  instance, 

\'ii4  =  4, 

while,  in  oilier  eases,  it  cannot  be  carried  out.     For  instance, 

a/J=?. 


12  ENGINEERING  MATHEMATICS. 

Attempting  to  calculate  yJ'2,  we  get, 

^2  =  1.4142135..  ., 

and  find,  no  matter  how  far  we  carry  the  calculation,  we  never 
come  to  an  end,  but  get  an  endless  decimal  fraction;  that  is, 
no  number  exists  in  our  system  of  numbers,  which  can  express 
-\2,  but  we  can  only  approximate  it,  and  carry  the  approxima- 
tion to  any  desired  degree ;  some  such  numbers,  as  7r,  have  been 
calculated  up  to  several  hundred  decimals. 

Such  numbers  as  \2,  which  cannot  be  expressed  in  any 
finite  form,  but  merely  approximated,  are  called  irrational 
numbers.  The  name  is  just  as  wrong  as  the  name  negative 
number,   or  imaginary   number.     There   is   nothing  irrational 

about  il'2.  If  we  draw  a  square,  with  1  foot  as  side,  the  length 
of  the  diagonal  is  -^2  feet,  and  the  length  of  the  diagonal  of 
a  square  obviously  is  just  as  rational  as  the  length  of  the  sides. 

Irrational  numbers  thus  are  those  real  and  existing  numbers, 
which  cannot  be  expressed  by  an  integer,  or  a  fraction  or  finite 
decimal  fraction,  but  give  an  endless  decimal  fraction,  which 
does  not  repeat. 

Endless  decimal  fractions  frequently  are  met  when  express- 
ing common  fractions  as  decimals.  These  decimal  representa- 
tions of  common  fractions,  however,  are  periodic  decimals, 
that  is,  the  numerical  values  periodically  repeat,  and  in  this 
respect  are  different  from  the  irrational  number,  .and  can,  due 
to  their  periodic    nature,  be   converted    into  a  finite   common 

fraction.     For  instance,  2.1387387 

Let 

x=       2.1387387 ; 

then, 

subl  racting, 

Hence, 


1000^  =  2138.7387387. 
999a; =2136.6 


2136.6  213(10  1187   77 
x=    -=  =-=-^z-  =  2-. 


999   9990   555   555 


io.  It  is 

since, 

but  it  also  is: 

since, 


THE  GENERAL  NUMBER.  13 

Quadrature  Numbers. 

V/-hT=(+2), 
(+2)x(+2)  =  (+4); 


^+4  =  (-2), 


(-2)X(-2)  =  (+4). 

Therefore,  \T+4  has  two  values,  (+2)  and  (—2),  and  in 
evolution  we  thus  first  strike  the  interesting  feature,  that  one 
and  the  same  operation,  with  the  same  numerical  values,  gives 
several  different  results. 

Since  all  the  positive  and  negative  numbers  are  used  up 
as  the  square  roots  of  positive  numbers,  the  question  arises, 
What  is  the  square  root  of  a  negative  number?  For  instance, 
"V  —4  cannot  be  —2,  as  —2  squared  gives  ■',  4,  nor  can  it  be  +2. 

-yJ^i  =  Sll  X  (  - 1)  =  ±  2  a/  -T,  and  the  question  thus  re- 
solves itself  into :  What  is  if^l  ? 

We  have  derived  the  absolute  numbers  from  experience, 
for  instance,  by  measuring  distances  on  a  line  Fig.  4,  from  a 
starting  point  A. 

-5     -4     -3     -2     -1       0     +1     +2     +3     +4     +5 

1 1 H ® 1 $ 1 ® 1 1 1 

C  A  B     _      v  . 

Fig.  4.     Negative  and  Positive  Numbers. 

Then  we  have  seen  that  we  get  the  same  distance  from  A, 
twice,  once  toward  the  right,  once  toward  the  left,  and  this 
has  led  to  the  subdivision  of  the  numbers  into  positive  and 
negative  numbers.  Choosing  the  positive  toward  the  right, 
in  Fig.  4,  the  negative  number  would  be  toward  the  left  (or 
inversely,  choosing  the  positive  toward  the  left,  would  give 
the  negative  toward  the  right). 

If  then  we  take  a  number,  as  +2,  which  represents  a  dis- 
tance AB,  and  multiply  by  (—1),  we  get  the  distance  AC=  —2 


14 


/•:  VGl  VEERING  MA  THEMA  T/<  'S. 


in  opposite  direction  from  A.     Inversely,  if  we  take  AC=  —2,. 
and  multiply  by  (— 1),  we  gel   AB     +  2;    that   is,  multiplica- 
tion by  (—1)  reverses  the  direction,  turns  it  through  180  (leg. 
If  we  multiply  +2  by  \    -1,  we  get  +2\    -1,  a   quantity 
of  which  we  do  not  yet  know  the  meaning.     Multiplying  once 

g<  it    +  2  X  \  "  ^X  \    rF=  -  2 ;    that   is , 


more 


by    V  —1,   we 


multiplying  a  number  +2,  twice  by  V  -1,  gives  a  rotation  of 
180  deg.,  and  multiplication  by  \  -1  thus  means  rotation  by 
half  of  180  (leg.;   or,  by  90  deg.,  and  +2\^:::I  thus  is  the  dis- 


-&- 


\RT 


OD 


x90° 


-e- 


Fig.  5. 


tance  in  the  direction  rotated  90  deg.  from  +2,  or  in  quadrature 

direction  AD  in  Fig.  5,  and  such  numbers  as  +2\  -1  thus 
are  quadrature  van/hers,  that  is,  represent  direction  not  toward 
the  right,  as  the  positive,  nor  toward  the  left,  as  the  negative 
numbers,  but  upward  or  downward. 

For  convenience  of  writing,    \    - 1   is  usually  denoted   by 

the  letter  j. 

ii.  Just  as  the  operation  of  subtraction  introduced  in  the 
negative  numbers  a  new  kind  of  numbers,  having  a  direction 
ISO  deg.  different,  that  is,  in  opposition  to  the  positive  num- 
bers, so  the  operation  of  evolution  introduces  in  the  quadrature 
number,  as  2/.  a  new  kind  of  number,  having  a  direction  90  deg. 


THE  GENERAL  NUMBER. 


15 


different;  that  is,  at  right  angles  to  the  positive  and  the  negative 
numbers,  as  illustrated  in  Fig.  6. 

As  seen,  mathematically  the  quadrature  number  is  just  as 
real  as  the  negative,  physically  sometimes  the  negative  number 
has  a  meaning — if  two  opposite  directions  exist — ;  sometimes  it 
has  no  meaning — where  one  direction  only  exists.  Thus  also 
the  quadrature  number  sometimes  has  a  physical  meaning,  in 
those  cases  where  four  directions  exist,  and  lias  no  meaning, 
in  those  physical  problems  where  only  two  directions    exist. 


+4/ 

+  3/ 
+2/ 


-4      -3 


1      0 


+  1 

-J 

V 

-3/ 


+  u 


-J 

Fig.  G. 


For  instance,  in  problems  dealing  with  plain  geometry,  as  in 
electrical  engineering  when  discussing  alternating  current 
vectors  in  the  plane,  the  quadrature  numbers  represent  the 
vertical,  the  ordinary  numbers  the  horizontal  direction,  and  then 
the  one  horizontal  direction  is  positive,  the  other  negative,  and 
in  the  same  manner  the  one  vertical  direction  is  positive,  the 
other  negative.  Usually  positive  is  chosen  to  the  right  and 
upward,  negative  to  the  left  and  downward,  as  indicated  in 
Fig.  G.  In  other  problems,  as  when  dealing  with  time,  which 
has  only  two  directions,  past  and  future,  the  quadrature  num- 
bers are  not  applicable,  but  only  the  positive  and  negative 


1G 


ENGINEERING  MA  THEMA  TICS. 


numbers.  In  still  other  problems,  as  when  dealing  with  illumi- 
nation, or  with  individuals,  the  negative  numbers  are  not 
applicable,  but  only  the  absolute  or  positive  numbers. 

Just  as  multiplication  by  the  negative  unit  (—  1)  means 
rotation  by  180  cleg.,  or  reverse  of  direction,  so  multiplication 
by  the  quadrature  unit,  j,  means  rotation  by  90  (leg.,  or  change 
from   the  horizontal  to  the  vertical   direction,  and   inversely. 


General    Numbers. 


12.  By  the  positive  and  negative  numbers,  all  the  points  of 
a  line  could  be  represented  numerically  as  distances  from  a 
chosen  point  A. 


Fig.  7.     Simple  Vector  Diagram. 

By  the  addition  of  the  quadrature  numbers,  all  points  of 
the  entire  plane  can  now  be  represented  as  distance's  from 
chosen  coordinate  axes  x  and  y,  that  is,  any  point  P  of  the 
plane,  Fig.  7,  has  a  horizontal  distance,  OB  =  +  3,  and  a 
vertical  distance,  BP=  +2j,  and  therefore  is  given  by  a 
combination  of  the  distances,  0B=  +3  and  BP  =  +'2j.  For 
convenience,  the  act  of  combining  two  such  distances  in  quad- 
rature with  each  other  can  be  expressed  by  the  plus  sign, 
and  the  result  of  combination  thereby  expressed  by  OB+BP 
=  3+2.,. 


THE  CEXERAL   XTMBER. 


Such  a  combination  of  an  ordinary  number  and  a  quadra- 
ture number  is  called  a  general  number  or  a  complex  quantity. 

The  quadrature  number  jb  thus  enormously  extends  the 
field  of  usefulness  of  algebra,  by  affording  a  numerical  repre- 
sentation of  two-dimensional  systems,  as  the  plane,  by  the 
general  number  a+jb.  They  are  especially  useful  and  impor- 
tant in  electrical  engineering,  as  most  problems  of  alternating 
currents  lead  to  vector  representations  in  the  plane,  and  there- 
fore can  be  represented  by  the  general  number  a+jb;  that  is, 
the  combination  of  the  ordinary  number  or  horizontal  distance 
a,  and  the  quadrature  number  or  vertical  distance  jb. 


Fig.  S.     Vector  Diagram. 

Analytically,  points  in  the  plane  are  represented  by  their 
two  coordinates:  the  horizontal  coordinate,  or  abscissa  x,  and 
the  vertical  coordinate,  or  ordinate  y.  Algebraically,  in  the 
general  number  a+jb  both  coordinates  are  combined,  a  being 
the  x  coordinate,  jb  the  y  coordinate. 

Thus  in  Fig.  8,  coordinates  of  the  points  are, 


Pi 


P 


4-3    y=-2, 

=   _9 


x=+3,     y=  +2 

x=  -3,     y=  +2  P4:    x=   -3     y 

and  the  points  are  located  in  the  plane  by  the  numbers: 
P1=3+2/    P2=3-2j     P3=-3+2/    P4  =  -3-2/ 


18  ENGINEERING   MATHEMATICS. 

13.  Since  already  the  square  rool  of  negative  numbers  has 
extended  the  system  of  numbers  by  giving  the  quadrature 
number,  the  question  arises  whether  still  further  extensions 
of  the  system  of  numbers  would  result  from  higher  roots  of 
negative  quantities. 

For  instance, 

4~r=? 

The  meaning  of   if— 1  we  find  in  the  same  manner  as  that 

of  v~T. 

A  positive  number  a  may  be  represented  on  the  horizontal 
axis  as  P. 

Multiplying  a  by  \' -1  gives  a-yj-l,  whose  meaning  we  do 
not  yet  know.  Multiplying  again  and  again  by  i  —1,  we  get,  after 
four  multiplications,  a(^'-l)4=  —a;  that  is,  in  four  steps  we 
have  been  carried  from  a  to    —a,  a  rotation  of  180  cleg.,  and 

iT-i  thus  means  a  rotation  of  ——  =  45  deg.,  therefore,  a\-l 

4 

is  the  point  1\  in  Fig.  9,  at  distance  a  from  the  coordinate 

center,  and   under  angle  45  deg.,  which  has  the  coordinates, 

x=         and  y=    —=j;  or,  is  represented  by  the  general  number, 
\  2  V2 

i+i 


Pi=a 


\  2 


\'-l,  however,  may  also  mean  a  rotation  by  135  deg.  to  P2, 
since  this,  repeated  four  times,  gives  4x135  =  540  deg., 
or  the  same  as  ISO  deg.,  or  it  may  mean  a  rotation  by  225  deg. 

or  by  315  deg.  Thus  four  points  exist,  which  represent  a  jj  — 1; 
the  points: 


'        \  2  '         \  2 

Therefore.  \  I  is  still  a  general  number,  consisting  of  an 
ordinary  and  a  quadrature  number,  and  thus  does  not  extend 
our  system  of  numbers  any  further. 


THE  GENERAL   NUMBER. 


19 


In  the  same  manner,  V  +  l  can  be  fount  I;  it  is  that  number, 
which,  multiplied  n  times  with  itself,  gives  +1.     Thus  it  repre- 

sents  a  rotation  by  -     -  cleg.,  or  any  multiple  thereof;    that  is, 

n 

300     ,  ,.  .  360 

the  x  coordinate  is  cos  qX- — ,  the    ?/   coordinate   sin   qX , 

n  n 

and, 

n/—                 300     .  .  360 

V  +1  =  cos  qX +  /  sin  qX  — , 

1       n      J        1       n 


when-  q  is  any  integer  number. 


Fig.  9.     Vector  Diagram  a -v  — 1. 


There  are  therefore  //  different  values  of  a\  +1,  which  lie 
equidistant  on  a  circle  with  radius  1,  as  shown  for  n  =  9  in 
Fig.  10. 

14.  In  the  operation  of  addition,  a  +  b  =  c,  the  problem  is, 
a  and  b  being  given,  to  find  c. 

The  terms  of  addition,  a  and  b,  are  interchangeable,  or 
equivalent,  thus:  a  +  b  =  b  +  a,  and  addition  therefore  has  only 
one  reverse  operation,  subtraction;  c  and  b  being  given,  a  is 
found,  thus:  a  =  c—b,  and  c  and  a  being  given,  b  is  found,  thus: 
b=c—a.      Either    leads    to    the    same    operation — subtraction. 

The    same    is    the    ease    in    multiplication;     aXb  =  c,     The 


20 


ENGINEERING  MA  THEMATICS. 


factors  a  and  b  arc  interchangeable  or  equivalent;    aXb  =  bXa 

c  c 

ami  the  reverse  operation,  division,  a  =  T  is  the  same  as  b=—. 

b  a 

In  involution,  however,   ab  =  c,   the  two  numbers  a  and   b 
are  not  interchangeable,  and  ab  is  not  equal  to  ba.     For  instance 
43  =  (>4and34  =  81. 

Therefore,  involution  has  two  reverse  operations: 

(a)  c  and  b  given,  a  to  be  found, 


or  evolution, 


a  —  \  c ; 


Fig.  10.     Points  Determined  by  V+l. 


(&)  c  and  a  given,  6  to  be  found, 

b  =  \oga  c; 


or,  logarithmation. 


Logarithmation. 


15.  Logarithmation  thus  is  one  of  the  reverse  operations 
of  involution,  and  the  logarithm  is  the  exponent  of  involution. 

Thus  a  logarithmic  expression  may  be  changed  to  an  ex- 
ponential, and  inversely,  and  the  laws  of  logarithmation  are 
the  laws,  which  the  exponents  obey  in  involution. 

1.  Powers  of  equal  base  are  multiplied  by  adding  the 
exponents:     abXan  =  ab+n.      Therefore,    the    logarithm    of    a 


THE  GENERAL  NUMBER.  21 

product  is  the  sum  of  the  logarithms  of  the  factors,  thus  loga-cXd 
=  logac  +  log„  d. 

2.  A  power  is  raised  to  a  power  by  multiplying  the  exponents: 
(aby>=abn. 

Therefore  the  logarithm  of  r*  power  is  the  exponent  times 
the  logarithm  of  the  base,  or,  the  number  under  the  logarithm 
is  raised  to  the  power  n,  by  multiplying  the  logarithm  by  n: 

loga  cn  =  n  loga  c, 

loga  1=0,  because  a0  =  1.  If  the  base  a  >  1,  loga  c  is  positive, 
if  c>l,  and  is  negative,  if  c<l,  but  >0.  The  reverse  is  the 
case,  if  a<l.  Thus,  the  logarithm  traverses  all  positive  and 
negative  values  for  the  positive  values  of  c,  and  the  logarithm 
of  a  negative  number  thus  can  be  neither  positive  nor  negative. 

loga  (— c)=loga  c+loga  (—  1),  and  the  question  of  finding 
the  logarithms  of  negative  numbers  thus  resolves  itself  into 
finding  the  value  of  loga  (  —  1). 

There  are  two  standard  systems  of  logarithms  one  with 
the  base  £  =  2.71828..  .*  and  the  other  with  the  base  10  is 
used,  the  former  in  algebraic,  the  latter  in  numerical  calcula- 
tions. Logarithms  of  any  base  a  can  easily  be  reduced  to  any 
other  base. 

For  instance,  to  reduce  6=loga  c  to  the  base  10:  6  =  logac 
means,  in  the  form  of  involution:  ab  =  c.  Taking  the  logarithm 
hereof  gives,  b  logio  a  =  logio  c,  hence, 

]ogiojc         _       ^       _  logio  c 
;_logi0a'     C         toa     ~ logio  a" 

Thus,  regarding  the  logarithms  of  negative  numbers,  we  need 
to  consider  only  logio  (  —1)  or  loge  (  —1). 

If  ]>  =  log£  (-D,  then  £**=  -1, 

and  since,  as  will  be  seen  in  Chapter  II, 

six  =  cos  x  +  j  sin  x, 
it  follows  that, 

cos  x+j  sin  x  =  —  1, 


*  Regarding  e,  see  Chapter  II,  p.  71. 


22  ENGINEERING  MA  THEMATICS. 

Hence,  x  =  ~,  or  an  odd  multiple  thereof,  and 

loge(-l)=M2tt+l), 

where  n  is  any  integer  number. 

Thus  logarithmation  also  leads  to  the  quadrature  number 
j,  but  to  no  further  extension  of  the  system  of  numbers. 

Quaternions. 

16.  Addition  and  subtraction,  multiplication  ami  division, 
involution  and  evolution  and  logarithmation  thus  represent  all 
the  algebraic  operations,  and  the  system  of  numbers  in  which 
all  these  operations  can  be  carried  out  under  all  conditions 
is  that  of  the  general  number,  a+jb,  comprising  the  ordinary 
number  a  and  the  quadrature  number  jb.  The  number  a  as 
well  as  b  may  be  positive  or  negative,  may  be  integer,  fraction 
or  irrational. 

Since  by  the  introduction  of  the  quadrature  number  jb, 
the  application  of  the  system  of  numbers  was  extended  from  the 
line,  or  more  general,  one-dimensional  quantity,  to  the  plane, 
or  the  two-dimensional  quantity,  the  question  arises,  whether 
the  system  of  numbers  could  be  still  further  extended,  into 
three  dimensions,  so  as  to  represent  space  geometry.  While 
in  electrical  engineering  most  problems  lead  only  to  plain 
figures,  vector  diagrams  in  the  plane,  occasionally  space  figures 
would  be  advantageous  if  they  could  be  expressed  algebra- 
ically. Especially  in  mechanics  this  would  be  of  importance 
when  dealing  with  forces  as  vectors  in  space. 

In  the  quaternion  calculus  methods  have  been  devised  to 
deal  with  space  problems.  The  quaternion  calculus,  however, 
has  not  yet  found  an  engineering  application  comparable  with 
that  of  the  general  number,  or,  as  it  is  frequently  called,  the 
complex  quantify.  The  reason  is  that  the  quaternion  is  not 
an  algebraic  quantity,  and  the  laws  of  algebra  do  not  uniformly 
apply  to  it. 

17.  With  the  rectangular  coordinate  system  in  the  plane. 
Fig.  II,  the  x  axis  may  represent  the  ordinary  numbers,  the  y 
axis  the  quadrature  numbers,  and  multiplication  by  j  -  \  -1 
represents  rotation  by  90  deg.     For  instance,  if  T\  is  a  point 


THE  GENERAL  NUMBER. 


23 


a+jb=3+2j,    the    point    P2,  90  deg.   away    from    1\,  would 
be: 

P2  =  ]Pl=]Xa  +  jb)=j(3+2j)=   -2+3/, 

To  extend  into  space,  we  have  to  add  the  third  or  z  axis, 
as  shown  in  perspective  in  Fig.  12.  Rotation  in  the  plane  xy, 
by  90  deg.,  in  the  direction  +x  to  +y,  then  means  multiplica- 
tion by  j.  In  the  same  manner,  rotation  in  the  yz  plane,  by 
90  deg.,  from  +y  to  +z,  would  be  represented  by  multiplica- 


>  + 


Fig.  11.     Vectors  in  a  Plane. 

tion  with  h,  and  rotation  by  90  deg.  in  the  zx  plane,  from  +z 
to  +x  would  be  presented  by  k,  as  indicated  in  Fig.  12. 

All  three  of  these  rotors,  j,  h,  k,  would  be  V^-l,  since  each, 
applied  twice,  reverses  the  direction,  that  is,  represents  multi- 
plication by  ( —  1). 

As  seen  in  Fig.  12,  starting  from  +x,  and  going  to  +y, 
then  to  +z,  and  then  to  +x,  means  successive  multiplication 
by  j,  h  and  k,  and  since  we  come  back  to  the  starting  point,  the 
total  operation  produces  no  change,  that  is,  represents  mul- 
tiplication by  (  +  1).     Hence,  it  must  be, 

jhk=  f  1. 


24 


ENGINEERING  MA T HEMATICS. 


Algebraically   this  is  not  possible,  since  each  of  the  three  quan- 
tities is    \    -1,   and    V -1  xV-1  xV  -1  =  - V^T,  and  not 

(  +  !)• 

+  y 


^ 


->+cc 


-V 
Fig.  12.     Vectors  in  Space,  jhk=  +1. 

If  we  now  proceed  again  from  x,  in  positive  rotation,  but 
first  turn  in  the  xz  plane,  we  reach  by  multiplication  with  k 
the  negative  z  axis,  —z,  as  seen  in  Fig.  13.     Further  multipli 

+  y 

i  i 


lica- 


-z 


k 


■>+« 


-y 


Fig.  13.     Vectors  in  Space,  khj=  —  1. 

tion  by  /'  brings  us  to  +y,  and  multiplication  by  j  to  —x,  and 
in  this  case  the  result  of    the  three  successive   rotations  bv 


THE  GENERAL  NUMBER.  25 

90  degv  in  the  same  direction  as  in  Fig.  12,   but   in  a  different 
order,  is  a  reverse;    that  is,  represents   (—1).     Therefore, 

khj=  -1, 
and  hence, 

jhk=  —klij. 

Thus,  in  vector  analysis  of  space,  we  see  that  the  fundamental 
law  of  algebra, 

aXb  =  bXa, 

does  not  apply,  and  the  order  of  the  factors  of  a  product  is 
not  immaterial,  but  by  changing  the  order  of  the  factors  of  the 
product  jhk,  its  sign  was  reversed.  Thus  common  factors 
cannot  be  canceled  as  in  algebra;  for  instance,  if  in  the  correct 
expression,  jhk  —  khj,\vo  should  cancel  by  j,  h  and  k,  as  could  be 
done  in  algebra,  we  would  get  + 1  =  —1,  which  is  obviously  wrong. 
For  this  reason  all  the  mechanisms  devised  for  vector  analysis 
in  space  have  proven  more  difficult  in  their  application,  and 
have  not  yet  been  used  to  any  great  extent  in  engineering 
practice. 

B.  ALGEBRA  OF  THE  GENERAL  NUMBER,  OR  COMPLEX 

QUANTITY. 

Rectangular  and  Polar  Coordinates. 

18.  The  general  number,  or  complex  quantity,  a+jb,  is 
the  most  general  expression  to  which  the  laws  of  algebra  apply. 
It  therefore  can  be  handled  in  the  same  manner  and  under 
the  same  rules  as  the  ordinary  number  of  elementary  arithmetic. 
The  only  feature  which  must  be  kept  in  mind  is  that  j2  =  —  1,  and 
where  in  multiplication  or  other  operations  j2  occurs,  it  is  re- 
placed by  its  value,  —  1.     Thus,  for  instance, 

(a+jb)(c  +  jd)  =ac+  jad  +  jbc  +  pbd 
=  ac  +  jad  +  jbc  —  bd 
=  (ac  —  bd)  +  j(ad + be). 

Herefrom  it  follows  that  all  the  higher    powers  of  j  can  be 

eliminated,  thus: 

/*=/,  p=  -1,  f=-y,  f=+l; 
f=+j,  f  =-T,  f=  -j,  ;'8=+l; 
/9=+j",  .  .  .    etc. 


21  i  ENGINEER  ING  M .  1  777  EM .  1  TICS. 

In  distinction  from  the  general  number  or  complex  quantity, 
the  ordinary  numbers,  +a  and  -a,  arc  occasionally  called 
>calars,  or  real  numbers.  The  general  number  thus  consists 
of  the  combination  of  a  scalar  or  real  number  and  a  quadrature 
number,  or  imaginary  number. 

Since  a  quadrature  number  cannot  be  equal  to  an  ordinary 
number  it  follows  that,  if  two  general  numbers  are  equal, 
their  real  components  or  ordinary  numbers,  as  well  as  their 
quadrature  numbers  or  imaginary  components  must  be  equal, 
thus,  if 

a+jb=c+jd, 

then, 

a  =  c    and     b  =  d. 

Every  equation  with  general  numbers  thus  can  be  resolved 
into  two  equations,  one  containing  only  the  ordinary  numbers, 
the  other  only  the  quadrature  numbers.     For  instance,  if 

x+jy  =  5-3j, 

then, 

x  =  5      and     y  =  —3. 

19.  The  best  way  of  getting  a  conception  of  the  general 
number,  and  the  algebraic  operations  with  it,  is  to  consider 
the  general  number  as  representing  a  point  in  the  plane.  Thus 
the  general  number  a+jb  =  Q+2.5j  may  be  considered  as 
representing  a  point  P,  in  Fig.  14,  which  has  the  horizontal 
distance    from    the    y  axis,   0A  =  BP  =  a  =  6,   and   the   vertical 

distance  from  the  x  axis,  OB  =  AP  =  6=2.5. 

The  total  distance  of  the  point  Pfrom  the  coordinate  center 
0  then  is 

or=V0A^  i  .1/'- 


=  Va2  +  62  =  VO2  +  2.52  =  6.5, 

and  the  angle,  which  this  distance  OP  makes  with  the  x  axis, 

is  given  by 

AP 

tan  0  =  = 

OA 

b     2.5 
=  — =0.417. 

a       (') 


THE  GENERAL  XTMIiER. 


27 


Instead   of  representing   the   general   number   by   the   two 

components,  a  and  b,  in  the  form  a+jb,  it  can  also  be  repre- 
sented by  the  two  quantities:  the  distance  of  the  point  P  from 
the  center  0, 


c  =  Va2+b2; 
and  the  angle  between  this  distance  and  the  x  axis, 


tan0=-. 

a 


-* — i — i— i — i — + 


Fig.  14.     Rectangular  and  Polar  Coordinates. 

Then  referring  to  Fig.   1  \. 

a  =  <■  cos  6     and     b  =  c  sin  6, 

and  the  general  number  a+jb  thus  can  also  be  written  in  the 
form, 

c(cos  6  +j  sin  d  i. 

The  form  a+jb  expresses  the  general  number  by  its 
rectangular  components  a  and  b,  and  corresponds  to  the  rect- 
angular coordinate's  of  analytic  geometry:  a  is  the  x  coordinate, 
b  the  y  coordinate. 

The  form  c(cos(9+/sm  6)  expresses  the  general  number  by 
what  may  be  called  its  polar  components,  the  radius  c  and  the 


L's  ENGINEERING  MATHEMATICS. 

angle  6,  and  corresponds  to  the  polar  coordinates  of  analytic 
geometry,  c  is  frequently  called  the  radius  vector  or  scalar, 
()  the  phase  angle  of  the  general  number. 

While  usually  the  rectangular  form  a+jb  is  more  con- 
venient, sometimes  the  polar  form  c(cos0  +/  sin  6)  is  preferable, 
and  transformation  from  one  form  to  the  other  therefore  fre- 
quently applied. 

Addition  and  Subtraction. 

20.  If  ai+jb1=C)+2.o  j  is  represented  by  the  point  Pi; 
this  point  is  reached  by  going  the  horizontal  distance  ai  =  6 
and  the  vertical  distance  &i=2.5.  If  a2+/&2=3+4/  is  repre- 
sented by  the  point  P2,  this  point  is  reached  by  going  the 
horizontal  distance  a2  =  3  and  the  vertical  distance  &2  =  4. 

The  sum  of  the  two  general  numbers  (aj  +/&i)  +  (a2+/62)  = 
(G +2.5/) +  (3 +4/),  then  is  given  by  point  P0,  which  is  reached 
by  going  a  horizontal  distance  equal  to  the  sum  of  the  hor- 
izontal distances  of  Pi  and  P2:  a0  =  cn  +a2  =  6+3  =  9,  and  a 
vertical  distance  equal  to  the  sum  of  the  vertical  distances  of 
Pi  and  P2:  &o=&i+&2=2.5+4  =  6.5,  hence,  is  given  by  the 
general  number 

a0+jb0=(al  +  a2)  +/(&i  +b2) 
=9+6.5/. 

Geometrically,  point  P0  is  derived  from  points  Pi  and  P2j 
by  the  diagonal  OPq  of  the  parallelogram  OPiP0P2,  constructed 

with  0P[  and  0P2  as  sides,  as  seen  in  Fig.  15. 

Herefrom  it  follows  that  addition  of  general  numbers 
represents  geometrical  combination  by  the  parallelogram  law. 

Inversely,  if  P0  represents  the  number 

ao+jbo  =  9+6.5/, 

and  Pi  represents  the  number 

a,  +/&]  =6  l  2.5/, 

the  difference  of  these  numbers  will  be  represented  by  a  point 
P 2,  which  is  reached  by  going  the  difference  of  the  horizontal 


the  ge vera  L  N I  rM HER 


29 


distances  and  of  the  vertical  distances  of  the   points  Pq  and 
P\.     1 *■>  thus  is  represented  by 


and 


a-2  =  ao  —  CL\  =  9  —  6  =  3, 
&2  =  &o-&i  =  6.5  -2.5  =  4. 


Therefore,  the  difference  of  the  two  general   numbers  (<xo+/&o) 
and  (g&i+j&i)  is  given  by  the  general  number: 


as  seen  in  Fig.  15. 


a2  +  j/>2  =  (ao  - a , )  +  ^  &0  -  /,  , ) 
=3+4/, 


H 1 1 1 1- 


Fig.  15.     Addition  and  Subtraction  of  Vectors. 

This  difference  a-2-\-jb2  is  represented  by  one  side  OP2  of 
the  parallelogram  OPiP0P2,  which  has  OPx  as  the  other  side, 

and  OPq  as  the  diagonal. 

Subtraction  of  general  numbers  thus  geometrically  represents 
the  resolution  of  a  vector  0P()  into  two  components  OP}  and 
0P2,  by  the  parallelogram  law. 

Herein  lies  the  main  advantage  of  the  use  of  the  general 
number  in  engineering  calculation :  If  the  vectors  are  represented 
by  general  numbers  (complex  quantities),  combination  and 
resolution  of  vectors  by  the  parallelogram  law  is  carried  out  by 


30  ENGINEERING   MATHEMATICS. 

simple  addition  or  subtraction  of  their  general  numerical  values, 
that  is,  by  the  simplest  operation  of  algebra. 

21.  General  numbers  are  usually  denoted  by  capitals,  and 
their  rectangular  components,  the  ordinary  number  and  the 
quadrature  number,  by  small  letters,  thus: 

A  =  a\  +  ja2 ; 

the  distance  of  the  point  which  represents  the  general  number  A 
from  the  coordinate  center  is  called  the  absolute  value,  radius 
or  scalar  of  the  general  number  or  complex  quantity.  It  is 
the  vector  a  in  the  polar  representation  of  the  general  number: 

A  =  a(cos  0  +j  sin  0), 


and  is  given  by  a=\/ar  +a22. 

The  absolute  value,  or  scalar,  of  the  general  number  is  usually 
also  denoted  by  small  letters,  but  sometimes  by  capitals,  and 
in  the  latter  case  it  is  distinguished  from  the  general  number  by 
using  a  different  type  for  the  latter,  or  underlining  or  dotting 
it,  thus: 

A  =  fli  +ja2\  or     A  =  a,  +  ja2 ;  ori=oi  +ja,2 

or  A  = «!  +  ja2 ;  or    A  =  ax  +  ./'"_> 


ft  =  \  fti2  +a22:     or     A  =  \  ax-  +a2J, 

an<  1  A  i  +  /ft2  =  ft  ( cos  d  -f  j  sin  0) ; 

or  ftT  +/o2  =  A(cos  0+j  sin  0). 

22.  The  absolute  value,  or  scalar,  of  a  general  number  is 
always  an  absolute  number,  and  positive,  that  is,  the  sign  of  the 
rectangular  component  is  represented  in  the  angle  0.  Thus 
referring  to  Fig.  16, 

A=ai+ja2  =  4:+Sj; 

gives,  a     \  fti2  +  a22  =  .'); 

tan0     |    =0.75; 

6  =  37  deg.: 
and  A      .">  (cos  37  deg.  ;  /sin  37  deg). 


THE  GENERAL  NUMBER. 


31 


The  expression 


gives 


A  =  d\  +/a2  =  4  —  3/ 


a=Vai2+a22  =  5; 


tan  6/  =  —  —  =  —  0.75 ; 
4 


0  =  -37  dog. ;     or      =  180  -37  =  143  de 


-4-3/        -4 


Fig.  16.     Representation  of  General  Numbers. 


Which  of  the  two  values  of  0  is  the  correct  one  is  seen  from 
the  condition  ai  =  acos#.  As  ax  is  positive,  +4,  it  follows 
that  cos  0  must  be  positive;  cos  (—37  deg.)  is  positive,  cos  143 
dee.  is  negative:  hence  the  former  value  is  correct : 

f 

A  =  5{cos(  -37  deg.)  +j  sin(  -37  deg.)} 
=  5(cos  37  deg.  —j  sin  37  deg.). 

Two  such  general  numbers  as  (4+3/)  and  (4-3/),  or, 
in  general, 

(a+jb)     and      (a-jb), 

are  called  conjugate   numbers.     Their   product  is  an    ordinary 
and  not  a  general  number,  thus:   (a+jb){a-jb)  =  a2 +b2. 


32  ENGINEERING  MATHEMATICS. 

The  expression 


A  =  ai+ja2=  -4+3/ 


gives 


a  =  v/ai2+a22  =  5; 

3 

tan0  =  ---=  -0.75; 
4 

#=-37  (leg.     or      =180-37  =  143  (leg.; 

but  since  a\=a  cos  0  is  negative,  —  4,  cos  6/  must  be  negative, 
hence,  0  =  143  (leg.  is  the  correct  value,  and 

-A=5(cos  143  deg.  +/  sin  143  cleg.) 
=  5(— cos  37  deg.  + /"sin  37  deg.) 

The  expression 

A  =  a\  +ja2  =  -4—3/ 

gives 


a  =  Vai2+a22  =  5; 

0  =  37  deg.;     or     =180+37  =  217  deg.; 

but  since  oi=o  cos  0  is  negative,   —4,  cos  0  must  be  negative, 
hence  0  =  217  deg.  is  the  correct  value,  and, 

A=5  (cos  217  deg.  +/  sin  217  deg.) 
=  5(  —  cos  37  deg.  — /  sin  37  (leg.) 

The  four  general  numbers,  +4+3],  +4—3/,  —4+3/,  and 
-4—3/,  have  the  same  absolute  value,  5,  and  in  their  repre- 
sentations as  points  in  a  plane  have  symmetrical  locations  in 
the  four  quadrants,  as  shown  in  Fig.  1(>. 

As  the  general  number  A  =  ai+/a2  finds  its  main  use  in 
representing  vectors  in  the  plane,  it  very  frequently. is  called 
a  vector  quantity,  and  the  algebra  of  the  general  number  is 
spoken  of  as  vector  analysis. 

Since  the  general  numbers  A  =  ai+ja2  can  0(!  made  to 
represent  the  points  of  a  plane,  they  also  may  be  called  plane 
numbers,  while  the  positive  and  negative  numbers,  +  a  and— a, 


THE  GENERAL   NUMBER.  33 

may  be  called  the  linear  numbers,  as  they  represent  the  points 
of  a  line. 

Example :  Steam  Path  in  a  Turbine. 

23.  As  an  example  of  a  simple  operation  with  ger>'  ral  num- 
bers one  may  calculate  the  steam  path  in  a  two-wheel  s;:t; •<■ 
of  an  impulse  steam  turbine. 


+2/ 


»»B>»- 


■>  +x 


Fig.  17.     Path  of  Steam  in  a  Two-wheel  Stage  of  an  Impulse  Turbine. 

Let  Fig.  17  represent  diagrammatically  a  tangential  section 
through  the  bucket  rings  of  the  turbine  wheels.  W\  and  W2 
are  the  two  revolving  wheels,  moving  in  the  direction  indicated 
by  the  arrows,  with  the  velocity  s  =  400  feet  per  sec.  /  are 
the  stationary  intermediate  buckets,  which  turn  the  exhaust 
steam  from  the  first  bucket  wheel  Wi,  back  into  the  direction 
required  to  impinge  on  the  second  bucket  wheel  W2.  The 
steam  jet  issues  from  the  expansion  nozzle  at  the  speed  Sn=2200 


;;i 


A'  VGINEERING  MATHEMATR  'S. 


feet  per  sec,  and  under  the  angle  0o  =  2O  deg.,  against  the  first 
bucket  wheel  W\. 

The  exhaust  angles  of  the  three  successive  rows  of  buckets, 
W\,  /,  and  W2,  are  respectively  24  deg.,  30  deg.  and  45  deg. 
These  angles  are  calculated  from  the  section  of  the  bucket 
exit  required  to  pass  the  steam  at  its  momentary  velocity, 
and  from  the  height  of  the  passage  required  to  give  no  steam 
eddies,  in  a  manner  which  is  of  no  interest  here. 

As  friction  coefficient  in  the  bucket  passages  may  be  assumed 
^  =  0.12;  that  is,  the  exit  velocity  is  1  —  Ay =0.88  of  the  entrance 
velocity  of  the  steam  in  the  buckets. 


>+& 


Fie.  IS.     Vector  Diagram  of  Velocities  of  Steam  in  Turbine. 

Choosing  then  as  x-axis  the  direction  of  the  tangential 
velocity  of  the  turbine  wheels,  as  y-axis  the  axial  direction, 
the  velocity  of  the  steam  supply  from  the  expansion  nozzle  is 
represented  in  Fig.  18  by  a  vector  OS0  of  length  n0  =  2200  feet 
per  sec.,  making  an  angle  00  =  20  deg.  witli  the  .r-axis:  hence, 
can   be  expressed  by  the  general  number  or  vector  quantity: 

§o  =  s0  (cos  0o+j  sindo) 

-2200  (cos  20  deg.  +j sin  20  deg.) 
=  2070  +  750/ ft.  per  sec. 

The  velocity  of  the  turbine  wheel  IT,  is  s  =  400  feet  per  second, 
and  represented  in  Fig.  18  by  the  vector  OS,  in  horizontal 
direction. 


THE  GENERAL   NUMBER.  35 

The  relative  velocity  with  which  the  steam  enters  the  bucket 
passage  of  the  first  turbine  wheel  \\\  thus  is: 

=  (2070  +  750/) -400 
=  1070  +  750/ ft.  per  sec. 

This  vector  is  shown  a-  0&\  in  Fig.  18. 
The  angle   Qv,   under  which  the  steam  enters   the    bucket 
passage  thus  is  given  by 

tan  0i  =  -[67q  =  0.450,    as    0i=24.3deg. 

This  angle  thus  has  to  be  given  to  the  front  edge  of  the 
buckets  of  the  turbine  wheel  TJY 

The  absolute  value  of  the  relative,  velocity  of  steam  jet 
and  turbine  wheel  W1}  at  the  entrance  into  the  bucket  passage. 
is 


Si  =  V16702+ 7502  =  1830  ft.  per  sec. 

In  traversing  the  bucket  passages  the  steam  velocity  de- 
creases by  friction  etc.,  from  the  entrance  value  Sj  to  the 
exit  value 

s2  =  s1(l-kf)  =  1830X0.88  =  1610  ft.  per  sec. 

and  since  the  exit  angle  of  the  bucket  passage  lias  been  chosen 
as  #2  =  24  deg.,  the  relative  velocity  with  which  the  steam 
leaves  the  first  bucket  wheel  Wx  is  represented  by  a  vector 
0S2  in  Fig.  IS,  of  length  .s2  =  1(310,  under  angle  24  deg.  The 
steam  leaves  the  first  wheel  in  backward  direction,  as  seen  in 
Fig.  17,  and  24  deg.  thus  is  the  angle  between  the  steam  jet 
and  the  negative  x-axis;  hence,  62  =  ISO  -24  =  150  deg.  is  the 
vector  angle.  The  relative  steam  velocity  at  the  exit  from 
wheel  Wx  can  thus  be  represented  by  the  vector  quantity 

No  =  s2(cos  d->  +/  sin  02) 

=  1610  (cos  156  deg.  +/  sin  156  deg.) 
=  -1470+055/. 

Since  the  velocity  of  the  turbine  wheel  lb,  is  >=400,  the 
velocity  of  the  steam  in  space,  after  leaving  the  first   turbine 


30  ENGINEERING  MATHEMATICS. 

wheel,  that  is,  the  velocity  with  which  the  steam    enters  the 
intermediate  /,  is 

=  (-1470 +655/) +400 
=  -1070+655/, 

and  is  represented  by  vector  OSs  in  Fig.  18. 
The  direction  of  this  steam  jet  is  given  by 

655 

tan  03=  -Jqtq^  -0.613, 

as 

03=  -31.G  deg.;     or,     180-31.6  =  148.4  deg. 

The  latter  value  is  correct,  as  cos  0S  is  negative,  and  sin  #3  is 
positive. 

The  steam  jet  thus  enters  the  intermediate  under  the  angle  ' 
of  148.4  deg.;  that  is,  the  angle  180-148.4  =  31.6  deg.  in  opposite 
direction.     The  buckets  of  the  intermediate  /  thus  must  be 
curved  in  reverse  direction  to  those  of  the  wheel  W\,  and  must 
be  given  the  angle  31.6  cleg,  at  their  front  edge. 

The  absolute  value  of  the  entrance  velocity  into  the  inter- 
mediate /  is 


s3  =  V10702  +  0552  =  1255  ft.  per  sec. 

In  passing  through  the  bucket  passages,  this  velocity  de- 
creases by  friction,  to  the  value : 

s4  =  s3(l -Ay)  =  1255X0.88  =  1105  ft.  per  sec, 

and  since  the  exit  edge  of  the  intermediate  is  given  the  angle: 
04  =  30  deg.,  the  exit  velocity  of  the  steam  from  the  intermediate 
is  represented  by  the  vector  0£4  in  Fig.  18,  of  length  s4  =  1105, 
and  angle  04=3O  deg.,  hence, 

»S4  =  H05  (cos  30  deg-  +  /  sin  30  deg.) 
=  955+550/  ft.  per  sec. 

This  is  the  velocity  with  which   the  steam    jet  impinges 

on  the  second  turbine  wheel  W2,  and  as  this  wheel  revolves 


THE  GENERAL  NUMBER.  37 

with  velocity  s=400,  the  relative  velocity,  that  is,  the  velocity 
with  which  the  steam  enters  the  bucket  passages  of  wheel  W2,  is, 

=  (955 +  550/) -400 
=  555  +  550/  ft.  per  sec; 


and  is  represented  by  vector  OS5  in  Fig.  18. 
The  direction  of  this  steam  jet  is  given  by 

550 
tan  6 5  =  ^W  =  0.990,     as    05  =  44.8  deg. 

Therefore,  the  entrance  edge  of  the  buckets  of  the  second 
wheel  ir_.  must  be  shaped  under  angle  05=44.8  deg. 
The  absolute  value  of  the  entrance  velocity  is 


S5  =  \/5552  +  5502  =  780  ft.  per  sec. 

In  traversing  the  bucket  passages,  the  velocity  drops  from 
the  entrance  value  S5,  to  the  exit  value, 

s6=s5(l  -Ay)  =780X0.88  =  690  ft.  per  sec. 

Since  the  exit  angles  of  the  buckets  of  wheel  W2  has  been 
chosen  as  45  deg.,  and  the  exit  is  in  backward  direction,  06  = 
180—45=135  deg.,  the  steam  jet  velocity  at  the  exit  of  the 
bucket  passages  of  the  last  wheel  is  given  by  the  general  number 

$5  =  s6(cos  06+  j  sin  06 ) 

=  690  (cos  135  deg.  +/  sin  135  deg.) 

=  -487+487/ ft,  per  sec, 


and  represented  by  vector  OS6  in  Fig.  18. 

Since  s  =  400  is  the  wheel  velocity,  the  velocity  of  the 
steam  after  leaving  the  last  wheel  W2,  that  is,  the  "lost" 
or  "  rejected  "  velocity,  is 

S7=S6+s 

=  (-487 +487/) +400 

=  -87  +  487/  ft,  per  sec, 
and  is  represented  by  vector  OS7  in  Fig.  18. 


38 


ENGINEERING  MA  THEM  A  Tit 'S. 


The  direction  of  the  exhaust  steam  is  given  by. 

487 


tan  07=  — 


5.6,    as    67  =  180  -80  =  100  deg., 


and  the  absolute  velocity  is, 


s7  =  V872+4872  =  495  ft.  per  sec 
Multiplication  of  General  Numbers. 

24.    If    A=ai+ja2   and    B  =  b\+jb2,    arc     two    general,    of 
plane  numbers,  their  product  is  given  by  multiplication,  thus: 

AB  =  (ai+ja2)(bi+jb2) 

=  ai&!  +jalb2  +ja2bi  +j2a2b2, 
and  since  j2  =  —  1, 

AB  =  (aibi  —a2b2)  +j(aib2  4-  a2bi), 

and  the  product  can  also  be  represented  in  the  plane,  by  a  point, 

C=ci+jc2, 


where, 
and 


ci=axbi  —a2b2, 

c2  =  aib2  +a2bi. 
For  instance,  A=2+j  multiplied  by  5  =  1+1.5/  gives 
ci  =2  XI -1X1.5=0.5, 

hence, 


c2  =  2Xl.5  +  lXl=4; 
C  =  0.5+4j, 


as  shown  in  Fig.  19. 

25.  The  geometrical  relation  between  the  factors  A  and  B 
and  the  product  C  is  better  shown  by  using  the  polar  expression; 
hence,  substituting, 


d\  =a  cos  a 

a2  =  a  sin  a 


and 


which  gives 


tan 


a=x/al2  +  a22 

0,1 


a 


"1 


and 


61- 

-b 

cos 

8 

&2= 

-b 
b 

sin 

=  V 

> 

V 

+b22 

b2 

tan 

ft 

"61 

77/ A'  GENERAL  NUMBER.  39 

the  quantities  may  be  written  thus: 

.4.  =a(cos  a  +j  sin  a); 

£  =  &(cos0+/sin  /?), 
and  then, 

C  =  AB  =  ab(voH  a+/sin  «)(cos  /?+  ,/'  sin  0) 

=  a6  {(cos  a  cos  V  — sin  a  sin  3)  +/(cos  a  sin  /?  -(-sin  a  cos  /?)] 
=  a&  {cos  (a  +/?)  +/  sin  (a  +/?)] : 


Fig.  19.     Multiplication  of  Vectors. 

that  is,  two  general  numbers  are  multiplied  by  multiplying  their 

absolute  values  or  vectors,  a  and  6,  and  adding  their  phase  angles 
a  and  /?. 

Thus,  to  multiply  the  vector  quantity,  A  =  ai+ja2  =  a  (cos 

<y  4- /  sin  a)by  B  =  bi  +jb2  =  b  (cos  (3+ j  sin  /?)  the  vector  0A  in  Fig. 
19,  which  represents  the  general  number  A,  is  increased  by  the 
factor  b  =  \Zbi2+b22,  and  rotated  by  the  angle  t3,  which  is  given 

bo 
bv  tan  t3  =  T-- 

1 
Thus,  a  complex  multiplier  B  turns  the  direction  of  the 

multiplicand  A,  by  the  phase  angle  of  the  multiplier  B,  and 

increases  the  absolute  value  or  vector  of  A,  by  the  absolute 

value  of  B  as  factor. 


40  ENGINEERING  MATHEMATICS. 

The  multiplier  B  is  occasionally  called  an  operator,  as  it 
carries  out  the  operation  of  rotating  the  direction  and  changing 
the  length  of  the  multiplicand. 

26.  In  multiplication,  division  and  other  algebraic  opera- 
tions with  the  representations  of  physical  quantities  (as  alter- 
nating currents,  voltages,  impedances,  etc.)  by  mathematical 
symbols,  whether  ordinary  numbers  or  general  numbers,  it 
is  necessary  to  consider  whether  the  result  of  the  algebraic 
ope  ration,  for  instance,  the  product  of  two  factors,  has  a 
physical  meaning,  and  if  it  has  a  physical  meaning,  whether 
this  meaning  is  such  that  the  product  can  be  represented  in 
the  same  diagram  as  the  factors. 

For  instance,  3X4  =  12;  but  3  horses  X  4  horses  does  not 
give  12  horses,  nor  12  horses2,  but  is  physically  meaningless. 
However,  3  ft,  X4  ft,  =  12  sq.ft.     Thus,  if  the  numbers  represent 

$ — 1 — 1 — 0  o    1 — 1 — 1 — 1 — 1 — 1 — i— e — 1 — 1 — 1 

O  A    B  C 

Fig.  20. 

horses,  multiplication  has  no  physical  meaning.  If  they  repre- 
sent feet,  the  product  of  multiplication  has  a  physical  meaning, 
but  a  meaning  which  differs  from  that  of  the  factors.  Thus, 
if  on  the  line  in  Fig.  20,  OA=3  feet,  OB =  4  feet,  the  product, 
12  square  feet,  while  it  has  a  physical  meaning,  cannot  be 
represented  any  more  by  a  point  on  the  same  line;  it  is  not 
the  point  OC  =  12,  because,  if  we  expressed  the  distances  OA 
and  OB  in  inches,  36  and  48  inches  respectively,  the  product 
would  be  30X48  =  1728  sq.in.,  while  the  distance  OC  would  be 
144  inches. 

27.  In  all  mathematical  operations  with  physical  quantities 
it  therefore  is  necessary  to  consider  at  every  step  of  the  mathe- 
matical operation,  whether  it  still  has  a  physical  meaning, 
and,  if  graphical  representation  is  resorted  to,  whether  the 
nature  of  the  physical  meaning  is  such  as  to  allow  graphical 
representation  in  the  same  diagram,  or  not. 

An  instance  of  this  general  limitation  of  the  application  of 
mathematics  to  physical  quantities  occurs  in  the  representation 
of  alternating  current  phenomena  by  general  numbers,  or 
complex  quantities. 


THE  GENERAL  NUMBER. 


41 


An  alternating  current  can  be  represented  by  a  vector  01 
in  a  polar  diagram,  Fig.  21,  in  which  one  complete  revolution 
or  360  dcg.  represents  the  time  of  one  complete  period  of  the 
alternating  current.  This  vector  01  can  be  represented  by  a 
general  number, 

I=ii+ji2, 

where  i\  is  the  horizontal,  1%  the  vertical  component  of  the 
current  vector  01. 


Fig.  21.     Current,  E.M.F.  and  Impedance  Vector  Diagram. 

In  the  same  manner  an  alternating  E.M.F.  of  the  same  fre- 
quency can  be  represented  by  a  vector  OE  in  the  same  Fig.  21, 
and  denoted  by  a  general  number, 

E  =  ei+je2. 

An  impedance  can  be  represented  by  a  general  number, 

Z  =  r-jx, 

where  r  is  the  resistance  and  x  the  reactance. 

If  now  we  have  two  impedances,  OZx  and  OZ2,  Zx=r\  —jx\ 
and  Z2  =  r2  —  jx2,  their  product  7jX  Z2  can  be  formed  mathema  - 
ically,  but  it  has  no  physical  meaning. 


42  ENGINEERING   MATHEMATICS. 

If  we  have  a  current  anda  voltage,  I=ii  +  ji2  and  E =ei+je2) 

the  product  of  current  and  voltage  is  the  power  P  of  the  alter- 
nating circuit. 

The  product  of  the  two  general  numbers  (  and  E  can  be 
formed  mathematically,  IE,  and  would  represent  a  point  C 
in  the  vector  plane  Fig.  21.  This  point  C,  however,  and  the 
mathematical  expression  IE,  which  represents  it,  does  not  give 
the  power  P  of  the  alternating  circuit,  since  the  power  P  is  not 
of  the  same-  frequency  as  /  and  E,  and  therefore  cannot  be 
represented  in  the  same  polar  diagram  Fig.  21,  which  represents 
/  and  E. 

If  we  have  a  current  /  and  an  impedance  Z}  in  Fig.  21; 
I=i\+ji2  and  Z  =  r—]x,  their  product  is  a  voltage,  and  as  the 
voltage  is  of  the  same  frequency  as  the  current,  it  can  be  repre- 
sented in  the  same  polar  diagram,  Fig.  21,  and  thus  is  given  by 
the  mathematical  product  of  I  and  Z, 

E=IZ=(i1+ji2){r-jx), 

=  (iir+i2x  )  +i(i-2r  —i\x). 

28.  Commonly,  in  the  denotation  of  graphical  diagrams  by 
general  numbers,  as  the  polar  diagram  of  alternating  currents, 
those  quantities,  which  are  vectors  in  the  polar  diagram,  as  the 
current,  voltage,  etc.,  are  represented  by  dotted  capitals:  E,  I, 
while  those  general  numbers,  as  the  impedance,  admittance,  etc., 
which  appear  as  operators,  that  is,  as  multipliers  of  one  vector, 
for  instance  the  current,  to  get  another  vector,  the  voltage,  are 
represented  algebraically  by  capitals  without  dot:  Z  =  r—jx  = 
impedance,  etc. 

This  limitation  of  calculation  with  the  mathematical  repre- 
sentation of  physical  quantities  must  constantly  be  kept  in 
mind  in  all  theoretical  investigations. 

Division  of  General  Numbers. 

20.  The  division  of  two  general  numbers,  A=aiA-ja2  and 
B=h+jb2,  gives, 

A     a\+j(i2 
•  ==  B  ~  61  +  jb-2 

This  fraction  contains  the  quadrature  number  in  the  numer- 
ator as  well  as  in  the  denominator.     The   quadrature   number 


77/ A'  GENERAL   NUMBER.  I: 

•  an  be  eliminated  from  fche  denominator  by  multiplying  numer 
ator  and  denominator  by  the  conjugate  quantity  of  the  denom- 
inator, l>\  —  jb2,  which  gives: 

r _  fai+,/Vr ?)(/;,  -jb2)     (axbi  +  a2b2)  +  j(a2b t  —  q3 62) 
•  "  (61  +jb2){lh  -jb2)~  bi2  +  b-22 

(i]b[  +a2J>2       .  n2b\  ~(i\b2 
'  6i2  +  622    +]     h2  +  b22    ' 


lor  instance, 


.4  _  6 +  2.5/ 
•      £   :  3+4/ 

_   (0  +  2.5/)(3-4/) 
:  (3+4/)(3-4/) 
28-16.5? 


25 

LI2-O.667. 


If  desired,  the  quadrature  number  may  be  eliminated  from 
the  numerator  and  left  in  the  denominator  by  multiplying  with 
the  conjugate  number  of  the  numerator,  thus: 

p_A     a\+ja2 
•" ~B~  h  •  jb2 

_(a]  +ja2)(ai  —ja2) 

(bi+jb2)(ai  -ja2) 

a{2+a22 


for  instance, 


C 


{a,\b\  +a2b2)  +j(aib2  —a2bi)' 
A     6+2.5/ 


B      3+4/ 

(fi  + 2.5 /)(()  -2.5/) 
(3  +  4/)(6-2.5/j 
42.25 


28  1  Hi. 5/ 


30.  Just  as  in  multiplication,  the  polar  representation  of 
the  general  number  in  division  is  more  perspicuous  than  any 
other. 


44  ENGINEERING  MATHEMATICS. 

Let  A  =  a(cos  a+j  sin  a)  be  divided  by  B  =  b(cos  ,8  +  j  sin  /3), 
thus: 

A     a(cos  a  +/  sin  a) 
•     5     6(cos/3+j  sin/3) 

a(cos  «  +  /  sin  a) (cos  /?  —  j  sin  /?) 
~6(cos  p+j  sin  /?)(cos  /?  -j  sin  /?) 

a |  (cos  <t  cos  /3  +  sin  a  sin  /?)  +/(sin  a  cos  /?  —cos  a  sin  /?) } 
=  6(cosa/3  +  sin*/3) 

=  ^ !  cos  («  -/?)  +/  sin  (a  -/?) } . 

That  is,  general  numbers  A  and  7?  are  divided  by  dividing 
their  vectors  or  absolute  values,  a  and  b,  and  subtracting  their 
phases  or  angles  a  and  /?. 

Involution  and  Evolution  of  General  Numbers. 

31.  Since  involution  is  multiple  multiplication,  and  evolu- 
tion is  involution  with  fractional  exponents,  both  can  be  resolved 
into  simple  expressions  by  using  the  polar  form  of  the  general 
number. 

If, 

A=(i\  +jao  =  a(cos  a+]  sin  a), 

then 

C=An=an(cos  na  +j  sin  no). 

For  instance,  if 

A  =  3  +4/ =  5  (cos  53  deg.  +j  sin  53  deg.); 
then, 

C  =  A4  =  54(cos  4X53  deg.  +/ sin  4x53  deg.) 

=  625(cos  212  deg.  +j  sin  212  deg.) 
=  G25(  -cos  32  deg.  -j  sin  32  deg.) 
=  625(  -0.848  -0.530?') 
=  -529-331  j. 

If,  A=ai+ja,2  =  a  (cos  a+j  sin  a),  then 

C  =  w  A  =  A'l=a"lcos-  +7  sin-) 

\       n     J        nl 

n/-(       «      .   .     a\ 

=  vo  I  cos  -    '   /  Sill  -  I. 
\       n     J         nl 


THE  GENERAL  NUMBER.  45 

32.  If,  in  the  polar  expression  of  A,  we  increase  the  phase 
angle  a  by  2tx,  or  by  any  multiple  of  2tt  :  Iqn,  where  q  is  any 
integer  number,  we  get  the  same  value  of  A,  thus: 

A  =  a\cos(a+2q7:)  +  /  sm(a+2q?z)\, 

since  the  cosine  and  sine  repeat  after  every  300  (leg,  or  2-. 
The  nth  root,  however,  is  different: 

~      nr-r      nr-(        a +2q7t      .   .     a  +2qn\ 
C=\/A  =  Va[cos-     ^-+7  sin-     -2-J. 
•  \  n         J  n     ) 

We  hereby  get  n  different  values  of  C,   for  q  =  0,  1,    2.  .  .n—  1; 
q  =  n  gives  again  the  same  as  q  =  0.     Since  it  gives 

a  +  2n;r     « 

=-  +  2?r; 

n         n 

that  is,  an  increase  of  the  phase  angle  by  300  deg.,  which  leaves 
cosine  and  sine  unchanged. 

Thus,  the  nth  root  of  any  general  number  has  n  different 
values,   and  these   values   have   the  same  vector  or  absolute 

n  —  2lZ 

term  Va,  but  differ  from  each  other  by  the  phase  angle  —  and 

its  multiples. 

For  instance,  let  A=  -529  -331/  =  025  (cos  212  deg.  + 
/  sin  212  deg.)  then, 

1/T      4/— -  /       212+360g  t  ,  .     212+300^ 
C=  \A=  -\025lcos-     —7 -  +  jsm-    —1 -) 

=  5 (cos  53  +  /  sin  53)  =  3  +  4/ 

=  5 (cos  143+/ sin  143)  =  5(  -cos  37+/ sin  37)  =  -4  +  3/ 
=  5(cos  233  +  /  sin  233)  =  5(  -cos  53 -  /  sin  53)  =  -3  - 4/ 
=  5(cos  323  +/  sin  323)  =  5(cos  37  -/  sin  37)     =  4  -3/ 
=  5 (cos  413+/ sin  413)  =  5 (cos  53+ /sin  53)     =3  +  4/ 


The  n  roots  of  a  general  number  A  =  a(cos  <*+/  sin  a)  differ 

'  2tz 
from  each  other  by  the  phase  angles  — ,  or  1/nth  of  300  deg., 

and  since  they  have  the  same  absolute  value  \  a,  it  follows,  that 
they  are  represented  by  n  equidistant  points  of  a  circle  with 
radius   \  a,  as  shown  in  Fig.  22,  for  ?2=4,  and  in  Fig.  23  for 


46 


ENGINEERING   MA  THEM  AT  K  'S. 


n  =  9.  Such  a  system  of  n  equal  vectors,  differing  in  phase  from 
each  other  by  1/n.th  of  360  deg.,  is  called  a  'polyphase  system,  or 
an  n-phase  system.  The  n  roots  of  the  general  number  thus 
give  an  n-phase  system. 

33-   For  instance,  vl  =  ? 

If  A=a  (cos  a  +  j  sin  a)  =  l.  this  means:  a  =  l,  «=0;   and 

hence, 


V\  = 


2q-      .   .     2f/7r 

cos h?  sin ; 

n  n 


P.=-4+3J 


Ps=-3-4j 


P,=3+4i 


P4=4-3J 


Fig.  22.     Roots  of  a  (ienoral  Number,  n  =  4. 
and  the  n  roots  of  the  unit  are 


q  =  Q 
2=1 


\   1  =  1 


360     .  .    360 
cos-     ■  +j  sin ; 


n 


H 


q=2 


o    300     •  •    o     360 
cos2x-     -Hsin2x — : 

n      J  n 


q  =  n  —  l 
However, 


1N360     .   .  ,    .360 

cos  (n  — 1)  -    -  +  ?  sm  (n-l)  — . 

n 


360 


n 


360 


360     .  .    300V 

cos  11  -    -  +  7  sin  q =  (  cos h  /  sm ; 

1    n       \         n      J  n    ' 


THE  CEXERAL   NUMBER. 


47 


hence,  the  n  roofs  of  1  arc. 


360     .  .    360V 
\  I      I  cos— -+jsin  — 1  , 


p 


n 


n 


where  q  may  be  any  integer  number. 

One  of  these  roots  is  real,  for  q=0,  and  is=  +1. 

If  n  is  odd,  all  the    other    roots  are   general,  or  complex 
numbers. 


n 


If  n  is  an  even  number,  a  second  root,  for  q=K,  is  also  real 
cos  180  +j  sin  180=  -1. 


Fig.  23.     Roots  of  a  General  Number,  n  =  9. 
If  n  is  divisible  by  4,  two  roots  are  quadrature  numbers,  and 
are    +],    for  q= j,  and  -/,  for  <?=-p 

34.  Using  the  rectangular  coordinate  expression  of  the 
general  number,  A  ~a>\.  +]«->,  the  calculation  of  the  roots  becomes 
more    complicated.     For    instance,    given  \CT  =  ? 

Let  C=4A=Cl+jc2; 

then,  squaring, 

A  =  (ci+fc2)2; 

hence, 

Ol  +jd2  =  (Ci2  -Co2)  +2JC\C,2. 

Since,  if  two  general  numbers  are  equal,  their  horizontal 
and  their  vertical  components  must  be  equal,  it  is: 

(i\=c\2-c22     and      a2  =  2c\c>. 


48  ENGINEERING   MATHEMATICS. 

Squaring  both  ('([nations  and  adding  them,  gives, 

ai2+a22  =  (ci2+c22)2. 
Hence : 


ci2+c22  =  vai2+a22, 

and  since  cr  —  c22  =  «i ; 


then,  cx2  =  h(Va12  +  a22+a1), 


and  c22  =  J  ( V  ax2  +  a22  -  a i ) . 

Thus 

=  |Ai\  «i2  +  a22  +  «i! 


and 
and 


Ci 


Co 


:\/])Val2  +  a22-a1\, 
<fA  =  fA {  Va{2  +  a2-  +  ai }  +  j'^/j  {  vV  +  a22  -«i  \ , 


which  is  a  rather  complicated  expression. 

35.  When  representing  physical  quantities  by  general 
numbers,  that  is,  complex  quantities,  at  the  end  of  the  calcula- 
tion the  final  result  usually  appears  also  as  a  general  number, 
or  as  a  complex  of  general  numbers,  and  then  has  to  be  reduced 
to  the  absolute  value  and  the  phase  angle  of  the  physical  quan- 
tity. This  is  most  conveniently  done  by  reducing  the  general 
numbers  to  their  polar  expressions.  For  instance,  if  the  result 
of  the  calculation  appears  in  the  form, 


d  _  (ai  +  ja2)(&i  +jb2)3\/ci  A-jc-2 

(d1+id2)2(e1+je2) 

by  substituting 

, — - -  a? 

a  =  Vai2+a22;     tan 


a  =  — 


b  =  Vbx2  +  b22\     tan/?  =  r2; 
and  so  on. 

p_o(cos  a  -fjsin  a)63(cos  /?+.f  sin  j9)3Vc(cos  y+j  sin  ?-)- 

d2(cos  d  +  j  sin  o)2e(cos  e+j  sin  e) 

alfiy/c 

—^-  \ cos(a+3,5  +  r/2  -2d  -  s)+j  sin  ( a+3/?+  r  2  -  2d  -  e) 


THE  GENERAL   NUMBER.  49 

Therefore,  the  absolute  value  of  a  fractional  expression  is 
the  product  of  the  absolute  values  of  the  factors  of  the  numer- 
ator, divided  by  the  product  of  the  absolute  values  of  the 
factors  of  the  denominator. 

The  phase  angle  of  a  fractional  expression  is  the  sum  of 
the  phase  angles  of  the  factors  of  the  numerator,  minus  the  sum 
of  the  phase  angles  of  the  factors  of  the  denominator. 

For  instance, 


(3  -4/)2(2  +  2/)^  -2.5  +(i/ 
5(4+3/)2v2 
25(cos307+/sin307)22\/2'(cos4.''>+/sin45)i/G7)(cosll4+ysinll4)i 

125(cos37+jsm37)2\/2 

114 

-2X37 


=  0.4^6.5 J  cosf2X307+ 45 


3 


+  j  sin  (2  X  307  4- 45  +-g— 2  X 37 j  f 


=  0.4^/(».5jcos  263 4- j  sin  2035 

=-0.740 }  -0.122  -0.092/!  =    -0.091  -0.74/. 

36.  As  will  be  seen  in  Chapter  II: 

K2     u3      u4 


.r-      .r4      x?      x* 


v»o  j*0  y7 

sin  x  =  x  —777-  +T^~  —j^r  + 
3       5       / 


Herefrom  follows,  by  substituting,  x  =  d,  u=jQ, 

cos  0+j  m\  0  =  e# 
and  the  polar  expression  of  the  complex  quantity, 

A  =a(cos  a  +j  sin  a  ), 
thus  can  also  be  written  in  the  form, 

A  =  aeia, 


50  ENGINEERING  MATHEMATICS. 

where   e  is  the  base  of  the  natural  logarithms, 

£  =  l  +  l+^  +  |4  +  A  +  -  •  .=2-71828  .  .  . 

1 1       Jo       1 4 

Since  any  number  a  can  be  expressed  as  a  power  of  any 
other  number,  one  can  substitute, 


and  the   general    number  thus    can 


a  = 

=  e% 

where 

a0=loge 

a- 

lOgio  OL 

logio  e' 

and 

the 

also  be  written 

in 

the  form 

j 

A  =  sai 

that  is  the  general  number,  or  complex  quantity,  can  be  expressed 
in  the  forms, 

A  =  ai+ja2 

m 

=  a(cos  a  +j  sin  a) 

The  last  two,  or  exponential  forms,  are  rarely  used,  as  they 
are  less  convenient  for  algebraic  operations.  They  are  of 
importance,  however,  since  solutions  of  differential  equations 
frequently  appear  in  this  form,  and  then  are  reduced  to  the 
polar  or  the  rectangular  form. 

37.  For  instance,  the  differential  equation  of  the  distribu- 
tion of  alternating  current  in  a  flat  conductor,  or  of  alternating 
magnetic  Mux  in  a  flat  sheet  of  iron,  has  the  form: 


(Py 
dr 


*=-2jc*y; 


and  is  integrate)  1  by  y  =  As    Vx,  where, 

F  =  V-2/c2  =  ±(l-:/)c; 
hence, 

y  =  Ax  e + &  ~  »cx  +  A2  e~ (1  - j)cx. 

This  expression,  reduced  to  the  polar  form,  is 

y  =  Ax£+cx(cos  ex  -j  sin  c.c)  +A2e~cx(cos  cx+j  sin  ex). 


THE  GENERAL  SIM I1HH.  51 

Logarithmation. 

38.  In  taking  the  logarithm  of  a  genera]  number,  the  ex- 
ponential expression  is  most  convenient,  thus : 

log£  (ai  +ja2)  =loge  a(cos  a  4-j  sin  a) 
=  log£a£}a 
=log£  a+logee,a! 
=log£a+/a; 

or,  if  6  =  base  of  the  logarithm,  for  instance,  b  =  10,  it  is: 

log6(ai  +ja2)  =log6  aeia  =log6  a+j'a  log6  e; 

or,  if  6  unequal  10,  reduced  to  logio; 

1         /  •    n      logio  a      .    login  e 

log6(a1+;a2)  =  -r^-7+i/«]   ' 


logio  b     J    logio  /> 


Note.  In  mathematics,  for  quadrature  unit  V—  1  is  always 
chosen  the  symbol  i.  Since,  however,  in  engineering  the  symbol  i 
is  universally  used  to  represent  electric  current,  for  the  quad- 
rature unit  the  symbol  j  has  been  chosen,  as  the  letter  nearest 
in  appearance  to  i,  and  j  thus  is  always  used  in  engineering 

calculations  to  denote  the  quadrature  unit  V  —  1. 


CHAPTER  II. 
POTENTIAL    SERIES    AND    EXPONENTIAL    FUNCTION. 

A.    GENERAL. 

39.  An  expression  such  as 

y-ih (1) 

represents  a  fraction;  that  is,  the  result  of  division,  and  like 
any  fraction  it  can  be  calculated;  that  is,  the  fractional  form 
eliminated,  by  dividing  the  numerator  by  the  denominator,  thus: 

l-x\l  =  l+x+x2+x3+.  .  . 

1-x 

+x 
x-x2 


+x2 
x2-x* 
+  x3. 

Hence,  the  fraction  (1)  can  also  be  expressed  in  the  form: 

y  =  T-—  =  l+x  +  x2  +  x:i  + (2) 

This  is  an  infinite  scries  of  successive  powers  of  x,  or  a  poten- 
tial series. 

In  the  same  manner,  by  dividing  through,    the  expression 

»-rb     (:i) 

can  be  reduced  to  the  infinite  series, 

?/  =  - =  l-x  +  x2-x3+  - (4) 

1  4- X 


POTENTIAL  SERIES  AND  EXPONENTIAL    FUNCTION.     53 

The  infinite  series  (2)  or  (4)  is  another  form  of  representa- 
tion of  the  expression  (1)  or  (3),  just  as  the  periodic  decimal 
fraction  is  another  representation  of  the  common  fraction 
(for  instance  0.0303.  .  .  .  =7/11). 

40.  As  the  series  contains  an  infinite  number  of  terms, 
in  calculating  numerical  values  from  such  a  series  perfect 
exactness  can  never  be  reached;  since  only  a  finite  number  of 
terms  are  calculated,  the  result  can  only  be  an  approximation. 
By  taking  a  sufficient  number  of  terms  of  the  series,  however, 
the  approximation  can  be  made  as  close  as  desired;  that  is, 
numerical  values  may  be  calculated  as  exactly  as  necessary, 
so  that  for  engineering  purposes  the  infinite  series  (2)  or  (4) 
gives  just  as  exact  numerical  values  as  calculation  by  a  finite 
expression  (1)  or  (2),  provided  a  sufficient  number  of  terms 
are  used.  In  most  engineering  calculations,  an  exactness  of 
0.1  per  cent  is  sufficient;  rarely  is  an  exactness  of  0.01  per  cent 
or  even  greater  required,  as  the  unavoidable  variations  in  the 
nature  of  the  materials  used  in  engineering  structures,  and  the 
accuracy  of  the  measuring  instruments  impose  a  limit  on  the 
exactness  of  the  result. 

For  the  value  x  =  0.5,  the  expression  (1)  gives  y  =  i — rr-  =  2; 

J.       U .  O 

while  its  representation  by  the  series  (2)  gives 

y= 1+0.5  +0.25  +0.125  +0.0625 +0.03125  +  .  .  .  (5) 

and  the  successive  approximations  of  the  numerical  values  of 
y  then  are: 

using  one  term:  '/=1  =1;  error:  —1 

"      two  terms:  y=  1  +  0.5  =1.5;  "  -0.5 

"      three  terms:  y=  1  +  0.5  +  0.-25  =1.75"  "  -0.25 

"      four  terms:  »/=  1  +  0.5  +  0.25  +  0.125  =1.875;        "  -0.125 

"      five  terms:  j/  =  l  +  0.5+0.25+0.125+0.0625  =  1.9375       "  -0.0625 

It  is  seen  that  the  successive  approximations  come  closer  and 
closer  to  the  correct  value,  y=2,  but  in  this  case  always  remain 
below  it;  that  is,  the  series  (2)  approaches  its  limit  from  below, 
as  shown  in  Fig.  24,  in  which  the  successive  approximations 
are  marked  by  crosses. 

For  the  value  x  =  0.5,  the  approach  of  the  successive 
approximations  to  the  limit  is  rat  her  slow,  and  to  get  an  accuracy 
of  0.1  per  cent,  that  is,  bring  the  error  down  to  less  than  0.002, 
requires  a  considerable  number  of  terms. 


54  ENGINEERING  MATHEMATICS. 

For  x  =  0.1  the  series  (2)  is 

y  =  l  +0.1  +0.01  +0.001  +0.0001  + (6) 

and  the  successive  approximations  thus  are 

1:  y  =  l; 

2:  y  =  l.l; 

3:  y  =  l.ll; 

4:  z/  =  l.lll; 

5:  r/  =  l.llll; 

and  as,  by  (1),  the  final  or  limiting  value  is 

1  10 

^=r=oT='9=1U11-- 


+  4  5 

+    3 

2 

+  y-r1 


Fig.  24.     Direct  Convergent  Series  with  One-sided  Approacli. 

the  fourth  approximation  already  brings  the  error  well  below 
0.1  per  cent,  and  sufficient  accuracy  thus  is  reached  for  most 
engineering  purposes  by  using  four  terms  of  the  series. 
41.     The  expression  (3)  gives,  for  .r  =  0.5,  the  value, 

Represented  by  series  (4),  it  gives 

y  =  1-0.5  +0.25  -0.125+0.0625-0.03125+  - (7) 

the  successive  approximations  are; 

1st:   y  =  l  =1;  error:  +0.333... 

2d:    ?/=l-0.5  ■  =0.5;  "  -0.1666... 

3d:    y=  1-0.5  +  0.25  =0.75;  "  +0.0S33... 

4th:  y- 1-0.5+0.25-0.125  =0.625;       "  -0.04166... 

5th:  y=l-0.5+().25-0.125+0.0625  =  0.6S75;  "  +0.020833... 


As  seen,  the  successive  approximations  of  tliis  series  come 
closer  and  closer  to  the  correct  value  ?y  =  0. (>(><>()  ....  but  in  this 
case  arc  alternately  above  and   below  the  correct   or  limiting 


POTENTIAL  SERIES  AND  EXPONENTIAL    FUNCTION.     55 

value,  that  is,  the  scries  (4  )  approaches  its  Limit  from  both  sides, 
as  shown  in  Fig.  25,  while  the  series  (2)  approached  the  Limit 
from  below,  and  still  other  series  may  approach  their  Limit 
from  above. 

With  such  alternating  approach  of  the  series  to  the  Limit, 
as  exhibited  by  series  (4),  the  limiting  or  final  value  is  between 
any  two  successive  approximations,  that  is,  the  error  of  any 
approximation  is  less  than  the  difference  between  this  and  the 
next  following  approximation. 

42.  Substituting  .r  =  2  into  the  expressions  (1)  and  (2), 
equation  (1)  gives 

1  - 


+ 
4 


G 


+ 
2 


Fig.  25.     Alternating  Convergent  Series, 
while  the  infinite  series  (2)  gives 

7y  =  l+2-r4+8  +  lG+32  +  ...; 
and  the  successive  approximations  of  the  latter  thus  are 
1;    3;     7;     15;    31;     63...; 

that  is,  the  successive  approximations  do  not  approach  closer 
and  closer  to  a  final  value,  but,  on  the  contrary,  get  further  and 
further  away  from  each  other,  and  give  entirely  wrong  results. 
They  give  increasing  positive  values,  which  apparently  approach 
00  for  the  entire  series,  while  the  correct  value  of  the  expression, 

by  (1),  is  y=  -1. 

Therefore,  for  x  =  2,  the  series  (2)  gives  unreasonable  results, 
and  thus  cannot  be  used  for  calculating  numerical  values. 

The  same  is  the  case  with  the  representation   (4)  of    the 
expression  (3)  for  x  =  2.     The  expression  (3)  gives 

1 


y=— -  =  0.333o 


1+2 


51  i  ENGINEERING  MA  THEM  A 1  'ICS. 

while  the  infinite  series  (4)  gives 

y= 1-2  +4  -8 +16  -32  + 

and  the  successive  approximations  of  the  latter  thus  are 

1;      -1;     +3;      -5;     +11;      -21;  .  .  .; 

hence,  while  the  successive  values  still  are  alternately  above 
and  below  the  correct  or  limiting  value,  they  do  not  approach 
it  with  increasing  closeness,  but  more  and  more  diverge  there- 
from. 

Such  a  series,  in  which  the  values  derived  by  the  calcula- 
tion of  more  and  more  terms  do  not  approach  a  final  value 
closer  and  closer,  is  called  divergent,  while  a  series  is  called 
convergent  if  the  successive  approximations  approach  a  final 
value  with  increasing  closeness. 

43-  While  a  finite  expression,  as  (1)  or  (3),  holds  good  for 
all  values  of  x,  and  numerical  values  of  it  can  be  calculated 
whatever  may  be  the  value  of  the  independent  variable  x,  an 
infinite  series,  as  (2)  and  (4),  frequently  does  not  give  a  finite 
result  for  every  value  of  x,  but  only  for  values  within  a  certain 
range.  For  instance,  in  the  above  series,  for  -1  <x<  +  l, 
the  series  is  convergent;  while  for  values  of  x  outside  of  this 
range  the  series  is  divergent  and  thus  useless. 

When  representing  an  expression  by  an  infinite  series, 
it  thus  is  necessary  to  determine  that  the  series  is  convergent; 
thai  is,  approaches  with  increasing  number  of  terms  a  finite 
limiting  value,  otherwise  the  series  cannot  be  used.  Where 
the  series  is  convergent  within  a  certain  range  of  x,  diver- 
gent outside  of  this  range,  it  can  be  used  only  in  the  range  oj 
convergency,  but  outside  of  this  range  it  cannot  be  used  for 
deriving  numerical  values,  but  some  other  form  of  representa- 
tion has  to  be  found  which  is  convergent. 

This  can  frequently  be  done,  and  the  expression  thus  repre- 
sented  by  one  series  in  one  range  and    by  another  series    in 

another  range.     For  instance,  the  expression  (1),  //  =  y-— ,  by 

substituting,  x  =  — ,  can  be  written  in  the  form 

1  u 

i+- 

u 


POTENTIAL  SERIES  AS  I)  EXPONENTIAL    FUNCTION.     57 

and  then  developed  into  a  scries   by  dividing  the  numerator 

by  the  denominator,  which  gives 

y  =  u  —  u2  +  u:i  —  a4  + .  .  . : 

or.  resubstituting  x, 

1111 

'-*-p+?-*+ ,Si 

which  is  convergent  for  .''  =  -,  and  for  x  =  2  it  gives 

y=0.5  -0.25  +0.125  -0.0625  +  .  .  .       (9) 
With  the  successive  approximations: 

0.5;     0.25:     0.375;     0.3125..., 

which  approach  the  final  limiting  value, 

2/ =0.333.  .. 

44.  An  infinite  series  can  be  used  only  if  it  is  convergent. 
Mathemetical  methods  exist  for  determining  whether  a  series 
is  convergent  or  not.  For  engineering  purposes,  however, 
these  methods  usually  are  unnecessary:  for  practical  use  it 
is  not  sufficient  that  a  series  be  convergent,  but  it  must  con- 
verge so  rapidly — that  is,  the  successive  terms  of  the  series 
must  decrease  at  such  a  great  rati — that  accurate  numerical 
results  are  derived  by  the  calculation  of  only  a  very  few  term.-; 
two  or  three,  or  perhaps  three  or  four.  This,  for  instance, 
is  the  case  with  the  series  (2)  and  (4)  for  z  =  0.1  or  less.  For 
x  =  0.5,  the  series  (2)  and  (4)  are  still  convergent,  as  seen  in 
(5)  and  (7),  but  are  useless  for  most  engineering  purposes,  as 
the  successive  terms  decrease  so  slowly  that  a  large  number 
of  terms  have  to  be  calculated  to  get  accurate  results,  and  for 
such  lengthy  calculations  there  is  no  time  in  engineering  work. 
If,  however,  the  successive  terms  of  a  series  decrease  at  such 
a  rapid  rate  that  all  but  the  first  few  terms  can  be  neglected, 
the  series  is  certain  to  be  convergent. 

In  a  series  therefore,  in  which  there  is  a  question  whether 
it  is  convergent  or  divergent,  as  for  instance  the  series 

11111  ,.  . 

w  =  l+-+-+T+^+,-+.  .  .  (divergent), 
^  l     3     -1      o     o 


58  ENGINEERING'   MATHEMATICS. 

or 

,11111 
2/=1-2+3_4+5_6"f""  (convergent)> 

the  matter  of  convergency  is  of  little  importance  for  engineer- 
ing calculation,  as  the  series  is  useless  in  any  case;  that  is,  docs 
not  give  accurate  numerical  results  with  a  reasonably  moderate 
amount  of  calculation. 

A  series,  to  be  usable  for  engineering  work,  must  have 
the  successive  terms  decreasing  at  a  very  rapid  rate,  and  if 
this  is  the  case,  the  series  is  convergent,  and  the  mathematical 
investigations  of  convergency  thus  usually  becomes  unnecessary 
in  engineering  work. 

45-  It  would  rarely  be  advantageous  to  develop  such  simple 
expressions  as  (1)  and  (3)  into  infinite  series,  such  as  (2)  and 
(4),  since  the  calculation  of  numerical  values  from  (1)  and  (3) 
is  simpler  than  from  the  series  (2)  and  (4),  even  though  very 
few  terms  of  the  series  need  to  be  used. 

The  use  of  the  series  (2)  or  (4)  instead  of  the  expressions 
(1)  and  (3)  therefore  is  advantageous  only  if  these  series  con- 
verge so  rapidly  that  only  the  first  two  terms  are  required 
for  numerical  calculation,  and  the  third  term  is  negligible; 
that  is,  for  very  small  values  of  x.  Thus,  for  £  =  0.01,  accord- 
ing to  (2), 

2/  =  l  +0.01+0.0001+...  =1+0.01, 

as  the  next  term,  0.0001,  is  already  less  than  0.01  per  cent  of 
the  value  of  the  total  expression. 

For  very  small  values  of  x,  therefore,  by  (1)  and  (2), 

y  =  T-  =  l+x, (10) 

and  by  (3)  and  (4), 

and  tnese  expressions  (10)  and  (11)  arc  useful  and  very  com- 
monly used  in  engineering  calculation  for  simplifying  work. 
For  instance,  if  1  plus  or  minus  a  very  small  quantity  appears 
as  factor  in  the  denominator  of  an  expression,  it  can  be  replaced 
by  1  minus  or  plus  the  same  small  quantity  as  factor  in  the 
numerator  of  the  expression,  and  inversely. 


POTENTIAL  SERIES  AND  EXPONENTIAL   FUNCTION.     59 

For  example,  if  a  direct-current  receiving  circuit,  of  resist- 
ance r,  is  fed  by  a  supply  voltage  c0  over  a  line  of  low 
resistance  tq,  what  is  the  voltage  e  at  the  receiving  circuit? 

The  total  resistance  is  r  +  rn\    hence,  the  current,  i  =  —  — , 

r+r0 

and  the  voltage  at  the  receiving  circuit  is 

e  =  ri  =  c^V^r (12) 

If  now  r0  is  small  compared  with  r,  it  is 

e=eQ-  —=e0\  1— ^\ (13) 

1+-°         l        '  j 
r 

As  the  next  term  of  the  series  would  be   (  — )  ,  the  error 

made  by  the  simpler  expression  (13)  is  less  than  (  — )  .  Thus, 
if  r0  is  3  per  cent  of  r,  which  is  a  fair  average  in  interior  light- 
ing circuits,  (  —  J  =0.032  =  0.0009,  or  less  than    0.1   percent; 

hence,  is  usually  negligible. 

46.  If  an  expression  in  its  finite  form  is  more  complicated 
and  thereby  less  convenient  for  numerical  calculation,  as  for 
instance  if  it  contains  roots,  development  into  an  infinite  series 
frequently  simplifies  the  calculation. 

Very  convenient  for  development  into  an  infinite  series 
of  powers  or  roots,  is  the  binomial  theorem, 


n(n-l)     ,     n(n-l)(n-2) 


(l±u)n  =  l±ni/+     -nr-     u?± ^-        -u4  +  . 

where 

|ra=lX2X3X.  .  .Xm. 


(14) 


Thus,    for    instance,    in    an    alternating-current    circuit    of 
resistance  r,  reactance  x,  and  supply  voltage  e,  the  current    is, 

i=    ,  6 (15) 

Vr2  +  x2 


60 


ENGINEERING  MA  THEM  AT  It 'S. 


If  this  circuit  is  practically  non-inductive,  as  an  incandescent 

lighting  circuit:    that  is.  if  x  is  small  compared  with  r,  (15) 

can  be  written  in  the  form, 

_  i 
e 


i\\-    -I- 


-inm 


(16) 


and  the  square  root  can  be  developed  by  the  binomial  (14),  thus, 
u={-J  ;  n=  — — ,  and  gives 


1  + 


2  v 


r\2     3/.r^4 


l--~^    It 


10  \r 


(17) 


In  this  series  (17),  if  .r  — O.lr  or  less;   that  is,  the  reactance 
is  not  more  than  10  per  cent  of  the  resistance,  the  third  term, 


o  ['-)  ,  is  less  than  0.01   per  cent;    hence,  negligible,  and  the 

scries  is  approximated  with  sufficient  exactness  by  the  first 
two  terms, 


[- 


1 

1    -77 


x  \- 


2  \r 


.     .     (18) 


and  equation  (16)  of  the  current  then  gives 


.--ili-i^Vl 


r 


ri    j 


(19) 


This  expression  is  simpler  for  numerical  calculations  than 
the  expression  (15),  as  it  contains  no  square  root. 

47.  Development  into  a  series  may  become  necessary,  if 
further  operations  have  to  be  carried  out  with  an  expression 
for  which  the  expression  is  not  suited,  or  at  least  not  well  suited. 
This  is  often  the  case  where  the  expression  has  to  be  integrated, 
since  very  few  expressions  can  be  integrated. 

Expressions  under  an  integral  sign  therefore  very  commonly 
have  to  be  developed  into  an  infinite  series  to  carry  out  the 
integration. 


POTENTIAL   SEMES   AND   EXPONENTIAL   FUNCTION.     61 


EXAMPLE    1. 
Of  the  equilateral  hyperbola  (Fig.  26), 

xy  =  a2,       .     . 


(20) 


the  length  L  of  the  arc   between    X\=2a  and  .r2  =  10a  is  to  bo 
calculated. 

An  elcnicnt  dl  of  the  are  is  (lie  hypothenuse  of  a  right  triangle 
with  dz  and  dy  as  cathetes.     It,  therefore,  is, 


dl=\/dx2+dy2 


Fig.  26.     Equilateral  Hyperbola. 


and  from  (20), 


a2 


y  =  —     and 


dy        a2 


=j    m 


hence,  the  length  L  of  I  he  arc,  from  X\  to  .rL>  is, 


*-£*-£ Ji+&* 


(21) 


d^ 

• 

Y/,r 

#?/= 

■a" 

03, 

^ 

- 

7  o«  •  •  •  • 

.r  eta         x2 

Substituting  (22)  in  (21)  gives. 


(22) 


(23) 


1 12  ENGINEERING  M.  1  THEM  A  TICS. 


x 
Substituting—  =  v;  that  is,  d.r  =  adv,  also  substituting 


2?i=—  =2     and     v>  =  —  =10, 
a  a 


gives 


L=aX  vi+^^ 

The  expression  under  the  integral  is  inconvenient  for  integra- 
tion; it  is  preferably  developed  into  an  infinite  series,  by  the 
binomial  theorem  (14). 

Write?/ =—:     and     n =-7,  then 


f      1        -    1       1         1  5 

+v±  2v*    8i<8  +  lQv12     128?;16  4 


and 


/""f         1        1  1  5  I  , 

[  1  1  1_ 

~ar\        2X3Xv4+7x8Xv8     11X16  Xv12 


i  1  , 


3X128  Xv16  Jw 

1/1        1\       1/1        1  \ 
i  o  vv     y23/      oo  uv     tv/ 


and  substituting  the  numerical  values, 

L  =  a{  (10-2) +-(0.125-0.001) 
o 


-7^(0.0078-0) +^(0.0001  -0) 
5b  1/0 


=  a{8  +0.0207  -  0.0001 }  =  8.0206a. 

As  seen,  in  this  series,  only  the  first  two  terms  are  appreciable 
in  value,  the  third  term  less  than  0.01  per  cent  of  the  total, 
and  hence  negligible,  therefore  the  series  converges  very 
rapidly,  and  numerical  values  can  easily  be  calculated  by  it. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     63 

For  x\  <2  a\  that  is,  ?'i  <2,  the  series  converges  less  rapidly, 
and  becomes  divergent  for  X\<a;  or,  ?'i<1.  Thus  this  scries 
(17)  is  convergent  for  r>  1,  but  near  this  limit  of  convergency 
it  is  of  no  use  for  engineering  calculation,  as  it  docs  not  converge 
with  sufficient  rapidity,  and  it  becomes  suitable  for  engineering 
calculation  only  when  n  approaches  2. 

EXAMPLE    2. 

48.  log  1=0,  and,  therefore  log  (l+.r)  is  a  small  quantity 
if  x  is  small,  log  (1+x)  shall  therefore  be  developed  in  such 
a  series  of  powers  of  x,  which  permits  its  rapid  calculation 
without  using  logarithm  tables. 

It  is 

.  fdu 

Iogu-J-; 

then,  substituting  (1+x)  for  u  gives, 

C  dx 
log  (1+x) -J  r-. (24) 

From  equation  (4) 
1 


=  l-z+x2-x3  +  . 

1+x 


•  1 


hence,  substituted  into  (24), 

log  (l+x)=  f(l  -.r+.r2-.r3+.  .  .)dx 

=  (dx  -  (xdx  +  (xHx  -  \x\lx  + .  .  . 

c2     c3     r4 
=  ^2+3-T+---         (25) 


x2  . 


hence,  if  x  is  very  small,   —  is  negligible,    and,  therefore,  all 

terms  beyond  the  first  are  negligible,  thus, 

log  (1  +x)  =x; 

while,  if  the  second  term  is  still  appreciable  in  value,  the  more 
complete,  but  still  fairly  simple  expression  can  be  used, 

x2       t       x 
log  (l+x)=x—  =  x(  1-., 


64  ENGINEERING   MATHEMATICS. 

If  instead  of  the  natural  logarithm,  as  used  above,  the 
decimal  logarithm  is  required,  the  following  relation  may  be 
applied : 

logio  a=logio£  log£  a  =  0.43 13  Iog£  a, 

logio  a  is  expressed  by  loge  a,   and  thus  (19),  (20)  (21)  assume 
the  form, 

logio  (1+x)  =0.4343^-^- +---+.  .  .  j; 
or,  approximately, 

logio(l+.r)=0.4343.r: 
or,  more  accurately, 

logio  d+.r)=0.4343.rM-|j 

B.    DIFFERENTIAL   EQUATIONS. 

49.  The  representation  by  an  infinite  series  is  of  special 
value  in  those  cases,  in  which  no  finite  expression  of  the  func- 
tion is  known,  as  for  instance,  if  the  relation  between  x  and  y 
is  given  by  a  differential  equation. 

Differential  equations  are  solved  by  separating  the  variables, 
that  is,  bringing  the  terms  containing  the  one  variable,  //,  on 
one  side  of  the  equation,  the  terms  with  the  other  variable  x 
on  the  other  side  of  the  equation,  and  then  separately  integrat- 
ing both  sides  of  the  equation.  Very  rarely,  however,  is  it 
possible  to  separate  the  variables  in  this  manner,  and  where 
it  cannot  be  done,  usually  no  systematic  method  of  solving  the 
differential  equation  exists,  but  this  has  to  be  done  by  trying 
different  functions,  until  one  is  found  which  satisfies  the 
e(iuatioi). 

In  electrical  engineering,  currents  and  voltages  are  dealt 
with  as  functions  of  time.  The  current  and  e.m.f.  giving  the 
power  lost  in  resistance  are  related  to  each  other  by  Ohm's 
law.  Current  also  produces  a  magnetic  field,  and  this  magnetic 
field  by  its  changes  generates  an  e.m.f.  —the  e.m.f.  of  self- 
inductance.  In  this  case,  e.m.f.  is  relate*  1  to  the  change  of 
current;  thai  is,  the  differential  coefficient  of  the  current,  and 
thus  also  to  the  differential  coefficient  of  e.m.f.,  since  the  e.m.f. 


POTENTIAL  SERIES   AND   EXPONENTIAL    FUNCTION,     65 

is  related  to  the  currenl  by  Ohm's  law.  In  ;i  condenser,  the 
current  and  therefore,  by  Ohm's  law,  the  e.m.f.,  depends  upon 
and  is  proportional  to  the  rate  of  change  of  the  e.m.f.  impressed 
upon  the  condenser;  that  is,  il  is  proportional  to  the  differential 
coefficient  of  e.m.f. 

Therefore,  in  circuits  having  resistance  and  inductance, 
or  resistance  and  capacity,  a  relation  exists  between  currents 
and  e.m.fs.,  and  their  differential  coefficients,  and  in  circuits 
having  resistance,  inductance  and  capacity,  a  double  relation 
of  this  kind  exists;  that  is,  a  relation  between  current  or  e.m.f. 
and  their  first  and  second  differential  coefficients. 

The  most  common  differential  equations  of  electrical  engineer- 
ing thus  are  the  relations  between  the  function  and  its  differential 
coefficient,  which  in  its  simplest  form  is, 

dx  =  y: (26) 

or 

%-<m (27» 

and  where  the  circuit  has  capacity  as  well  as  inductance,  the 
second  differential  coefficient  also  enters,  and  the  relation  in 
its  simplest  form  is, 

%=y> <*> 

or 

%-<*> (29) 

and  the  most  general  form  of  this  most  common  differential 
equation  of  electrical  engineering  then  is, 

Pi+2c^+ay+b  =  0 (30) 

ax2        ax 

The  differential  equations  (20)  and  (27)  can  easily  be  inte- 
grated by  separating  the  variables,  but  not  so  with  equations 
(28),  (29)  and  (30);   the  latter  are  preferably  solved  by  trial. 

50.  The  general  method  of  solution  may  be  illustrated  with 
the  equation  (26): 

dy    y (26) 


dx 


66  ENGINEERING   MA  THEMATIC 'S. 

To  determine  whether  this  equation  can  be  integrated  by  an 
infinite  series,  choose  such  an  infinite  series,  and  then,  by  sub- 
stituting it  into  equation  (26),  ascertain  whether  it  satisfies 
the  equation  (26);  that  is,  makes  the  left  side  equal  to  the  right 
side  for  every  value  of  .r. 

Let, 

;y  =  ao+ai£  +  a2-r2+a,>r3  +  tt4.r4  + (31) 

be  an  infinite  series,  of  which  the  coefficients  a0,  ci\,  a2,  a3.  .  . 
are  still  unknown,  and  by  substituting  (31)  into  the  differential 
equation  (26),  determine  whether  such  values  of  these  coefficients 
can  be  found,  which  make  the  series  (31)  satisfy  the  equation  (26). 
Differentiating  (31)  gives, 

~  =  ai+2a2x  +  3a3a-2+4a4.r3  + (32) 

ax 

The  differential  equation  (26)  transposed  gives, 

i-»-° (33) 

Substituting  (31)  and  (32)  into  (33),  and  arranging  the  terms 
in  the  order  of  x,  gives, 

(ai  —  ao)  +  (2ao  —  a\)x  4-  (3a3  —  a2)x2 

+  (4a4~a3)x3  +  (5a5-ai)x4  +   .  .=0.     .     (34) 

If  then  the  above  series  (31)  is  a  solution  of  the  differential 
equation  (26),  the  expression  (34)  must  be  an  identity;  that  is, 
must  hold  for  every  value  of  x. 

If,  however,  it  holds  for  every  value  of  x,  it  does  so  also 
for  x  =  0,  and  in  this  case,  all  the  terms  except  the  first  vanish, 
and  (34)  becomes, 

a,\  —  ao=0;     or,     ai=cto (35) 

To  make  (31)  a  solution  of  the  differential  equation  («i-a0) 
must  therefore  equal  0.  This  being  the  case,  the  term  {a\  —  ao) 
can  be  dropped  in  (34),  which  then  becomes, 

(2a 2 - ai)x  +  (3o3 -  a2)x2  +  (4a4-  a3)z3  +  (5a5 -  a4)x*  + . . .  =0; 

or, 

x\(2a2-al)  +  (3a3-a2)x  +  (4a,i-a^x2  +  .  .  .}=0. 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     67 


Since  this  equation  must  hold  for  every  value  of  x,  the  second 
term  of  the  equation  must  be  zero,  since  the  first  term,  x,  is 
not  necessarily  zero.     This  gives, 

(2a2-  «i )  +  (3a3  -  a2)x  +  (4a  t -  a3 >.<  -  ;  .  .  .     0. 

As  this  c(|uation  holds  for  every  value  of  x,  it  holds  also  for 
x  =  0.  In  this  case,  however,  all  terms  except  the  first  vanish, 
and, 

2«2-ai=0; (36) 

hence, 

"  i 


and  from  (35), 


02  =  T7- 


Continuing  the  .same  reasoning, 

3(7,3  —  «2  =  0,     4a4  —  a-3  =  0,  vie. 

Therefore,  if  an  expression  of  successive  powers  of  x,  such  as 
(34),  is  an  identity,  that  is,  holds  for  ever  if  value  of  x,  then  all 
the  coefficients  of  all  the  powers  of  x  must  separately  be  zero* 

Hence, 

ai  — ao  =  0;    or    ai  =  a0; 


2a2—ai=0;     or 


a\     a0 


3a3  — «2  =  0;     or     a3  =  ^-=pr, 
4a4  —  «3  =  0 ;     or    a4  =  -j  =  jj ; 


etc.. 


etc. 


(37) 


*  The  reader  must  realize  the  difference  between  an  expression  (34),  as 
equation  in  x,  and  as  substitution  product  of  a  function;  that  is,  as  an 
identity. 

Regardless  of  the  values  of  the  coefficients,  an  expression  (34)  as  equation 
gives  a  number  of  separate  values  of  x,  the  roots  of  the  equation,  which 
make  the  left  side  of  (34)  equal  zero,  that  is,  solve  the  equation.  If,  however, 
the  infinite  series  (31)  is  a  solution  of  the  differential  equation  (26),  then 
the  expression  (34),  which  is  the  result  of  substituting  (31)  into  (26),  must 
be  correct  not  only  for  a  limited  number  of  values  of  x,  which  arc  the  roots 
of  the  equation,  but  for  all  values  of  xu  that  is,  no  matter  what  value  is 
chosen  for  x,  the  left  side  of  (34)  must  always  give  the  same  result,  0,  that 
is,  it  must  not  be  changed  by  a  change  of  x,  or  in  other  words,  it  must  not 
contain  x,  hence  all  the  coefficients  of  the  powers  of .;:  must,  be  zero. 


68 


ENGINEERING   MATHEMA  Til 'S. 


Therefore,  if  the  coefficients  of  the  series  (31)  are  chosen 
by  equation  (37),  this  series  satisfies  the  differential  equation 
(26);   that  is. 


.r2     .r3     x4  1 


y  =  a()<  1+-c+-2+[3+rj  +  -  •  •  j 

is  the  solution  of  the  differential  equation, 

dy 


(38) 


dx 


=y- 


51.   In    tin1   same    manner,    the    differential   equation    (27), 

AZ  (30) 


=  (12 

dx         '  ' 

is  solved  by  an  infinite  series, 

2  =  ar,+aix  +  a2x2  Jrasx's+.  . .,    ....     (40) 

and   the  coefficients  of  this  series  determined   by  substituting 
(40)  into  (39),  in  the  same  manner  as  done  above.     This  gives, 

(01—  aa0)  +  (2a2  —  aa\)x  +  (3a3  —  aao)x2 

+  (4a4-at/3).r3  +  ...=0,  .     (41) 

and,  as  this  equation  must  be  an  identity,  all  its  coefficients 
must  be  zero;  that  is, 

a\  —  aa0  =  0;     or     ai=aao; 

a         a2 

2a>— aai  =  0;     or     a2  =  «i  :j  =  ao  Ty) 

a      '  a3 
3a3  —  aa-2  =  0 ;     or     a3  =  a2  tt  =  ao  77 ; 

o  o 

a         a4 
4a4— aa3  =  0;     or     a4  =  a37  =  aorr; 

etc.,  etc. 

and  the  solution  of  differential  equation  (39)  is, 


(42) 


2  =  a0\  1  +ax-{ — -r- 


e-'.r-     a3.r3     a4.r4 


& 


+  .  . 


(43) 


52.  These  solutions,  (38)  and  (43),  of  the  differential  equa- 
tions (26)  and  (39),  are  not  single  solutions,  but  each  contains 
an   infinite   number  of  solutions,   as  it   contains   an   arbitrary 


POTENTIAL   SERIES   AND  EXPONENTIAL   FUNCTION.     69 

constant  a0;    that  is.  a  constant  which  may  have  any  desired 
numerical  value 

This  can  easily  be  seen,  since,  if  z  is  a  solution  of  the  dif- 
ferential equation, 

dz 

dx  =  aZ> 

then,  any  multiple,  or  fraction  of  z,  bz,  also  is  a  solution  of  the 
differential  equation; 

d(bz) 

since  the  b  cancels. 

Such  a  constant,  a(),  which  is  not  determined  by  the  coeffi- 
cients of  the  mathematical  problem,  hut  is  left  arbitrary,  ami 
requires  for  its  determinations  some  further  condition  in 
addition  to  the  differential  equation,  is  called  an  integration 
constant.  It  usually  is  determined  by  some  additional  require- 
ments of  the  physical  problem,  which  the  differential  equation 
represents;  that  is,  by  a  so-called  terminal  condition,  as,  for 
instance,  by  having  the  value  of  y  given  for  some  particular 
value  of  r,  usually  for  x=0,  or  z=oo. 

The  differential  equation, 

t-r. w 

thus,  is  solved  by  the  function. 

//    "ii.'/.i (45) 

where, 

Jfo-1+s+f+l+g-1 (46) 


and  the  differential  equation, 

dz 


=  az, (47) 

dx 


is  solve*  1  by  the  function, 

z-o-qZo, (48) 

where, 

z0  =  l+ax+-^-+-^-+    j        (49J 


70  ENGINEERING  MATHEMATICS. 

2/o  and  z0  thus  are  the  simplest  forms  of  the  solutions  y  and  z 
of  the  differential  equations  (26)  and  (39). 

53.  It  is  interesting  now  to  determine  the  value  of  yn.  To 
raise  the  infinite  series  (46),  which  represents  y0,  to  the  nth 
power,  would  obviously  be  a  very  complicated  operation. 

However, 

diin  ,  dy 

dii 
and  since  from  (44)  w~  =  ^> (5-0 

by  substituting  (51)  into  (50), 

dyn 

hence,  the  same  equation  as  (47),  but  with  yn  instead  of  2. 
Hence,  if  y  is  the  solution  of  the  differential  equation, 

dy 

dx  =  y' 

then  z  =  yn  is  the  solution  of  the  differential  equation   (52), 

dz 
~r=nz. 

ax 

However,  the  solution  of  this  differential  equation  from  (47), 
(48),  and  (49),  is 

z  =  a0zo; 

n2x2    n3x3 

z0  =  l+nx+— -t^—  +. ..  ; 


that  is,  if 

y0  =  l+x+:^+^+...) 


X2      £3 


then, 

Zo  =  yon  =  1+n*+11^-+:j^  +  -  •  .  ;   •     .     .     (53) 


n2x2    n3xs 


therefore  the  series  y  is  raised  to  the  nth  power  by  multiply- 
ing the  variable  x  by  n. 


POTENTIAL  SERIES  AND  EXPONENTIAL   E UNCTION.     71 

Substituting  now  in  equation  (53)  for  n  the  value  -  gives 

1  111 

3fo*=l+l+9+j3+J4+...  J      ....     (54) 

that  is,  a    constant    numerical  value.      This  numerical  value 
equals  2.7182818.  .  .,  and  is  usually  represented  by  the  symbol  e. 
Therefore, 

hence, 

x2     /■''    ./■' 
y0  =  e*  =  l+X+—  +y^+U  +  -  ■  ■  , (55) 

and 

.    .  ,  n'-x-     n3x3     n4x4  ,     s 

2o  =  2/0n=(^)n=^x  =  l+nx+-^-+-^-+-jJ-  +  ...  ;     (50) 

therefore,  the  infinite  series,  which  integrates  above  differential 
equation,  is  an  exponential  function  with  the  base 

£  =  2.7182818 (57) 

The  solution  of  the  differential  equation, 

dii 

i=y> ^ 

thus  is, 

y=%ex, (59) 

and  the  solution  of  the  differential  equation, 

dy 

is, 

V  =  a0eax, (61) 

where  a0  is  an  integration  constant. 

The  exponential  function  thus  is  one  of  the  most  common 
functions  met  in  electrical  engineering  problems. 

The  above  described  method  of  solving  a  problem,  by  assum- 
ing a  solution  in  a  form  containing  a  number  of  unknown 
coefficients,  a0,  a,\,  a^ . . .,  substituting  the  solution  in  the  problem 
and  thereby  determining  the  coefficients,  is  called  the  method 
of  indeterminate  coefficients.     It  is  one  of  the  most  convenient 


72  ENGINEERING  MA  THEM  A  TK  'S. 

and    most    frequently    used    methods    of     solving   engineering 
problems. 

EXAMPLE    1. 

54.  In  a  4-pole  500-volt  50-kw.  direct-current  shunt  motor, 
the  resistance  of  the  field  circuit,  inclusive  of  field  rheostat,  is 
250  ohms.  Each  field  pole  contains  4000  turns,  and  produces 
at  500  volts  impressed  upon  the  field  circuit,  8  mcgalines  of 
magnetic  flux  per  pole. 

What  is  the  equation  of  the  field  current,  and  how  much 
time  after  closing  the  field  switch  is  required  for  the  field  cur- 
rent to  reach  00  per  cent  of  its  final  value? 

Let  r  be  the  resistance  of  the  field  circuit,  L  the  inductance 
of  the  field  circuit,  and  i  the  field  current,  then  the  voltage 
consumed  in  resistance  is, 

er  =  ri. 

In  general,  in  an  electric  circuit,  the  current  produces  a 
magnetic  field;  that  is,  lines  of  magnetic  flux  surrounding  the 
conductor  of  the  current;  or,  it  is  usually  expressed,  interlinked 
with  the  current.  This  magnetic  field  changes  with  a  change  of 
the  current,  and  usually  is  proportional  thereto.  A  change 
of  the  magnetic  field  surrounding  a  conductor,  however,  gen- 
erates an  e.m.f.  in  the  conductor,  and  this  e.m.f.  is  proportional 
to  the  rate  of  change  of  the  magnetic  field;  hence,  is  pro- 
portional   to    the    rate    of    change    of    the    current,    or    to 

di 

— ,  with  a  proportionality  factor  L,  which  is  called  the  induct- 

(Jiv 

ance  of  the  circuit.     This  counter-generated  e.m.f.  is  in  oppo- 

•  •  i  t  di         ,     ,  , 

sition  to  the  current,    —L-,   and  thus  consumes  an   e.ni.l., 

di 

+  L-T.)  which  is  called  the  e.m.f.  consumed  by  self-inductance, 

tits 

or  inductance  e.m.f. 

Therefore,  by  the  inductance,  L,  of  the  field  circuit,  a  voltage1 
is  consumed  which  is  proportional  to  the  rate  of  change  of  the 
field  current,  thus, 

e--Ldi- 


POTENTIAL  SERIES  AND  EXPONENTIAL    FUNCTION.     73 

Since  the  supply  voltage,  and  thus  the  total  voltage  consumed 
in  the  field  circuit,  is  e  =  500  volts, 

r  di 
e=n  +  Ljt; (62) 

or,  rearranged, 

di     e—ri 

Substituting  herein, 

u  =  e— ri\ (63) 

hence, 

(hi  di 

gives, 


di  =      r  dV 


^  =  -r-n  «>4) 

dt         L l 

r 

This   is  the   same   differential  equation  as  (39),   with   «=—-=, 

Li 

and  therefore  is  integrated  by  the  function, 

U  =  (l,)S      L,     ; 

therefore,  resubstituting  from  (63), 


e—n  =  a{):    L    , 


and 


r 


•  e  «0     ~Ll  /Prx 

1= £  (65) 

r      r 


This  solution  (65),  still  contains  the  unknown  quantity  c/0: 
or,  the  integration  constant,  and  this  is  determined  by  know- 
ing the  current  i  for  some  particular  value  of  the  time  /. 

Before  closing  the  field  switch  and  thereby  impressing  the 
voltage  on  the  field,  the  field  current  obviously  is  zero.  In  the 
moment  of  closing  the  field  switch,  the  current  thus  is  still 
zero;    that  is. 

i    0  for  t  =  0.  .     (66) 


74  ENGINEERING   MATHEMATICS. 

Substituting  these  values  in  (65)  gives, 


a     e     0() 

0  = —:     or     a0  =  +e, 

r      r 


hence, 


r 


i  =  ~[l-£"Lt 


is  the  final  solution  of  the  differential  equation  (62);  Jiat  is, 
it  is  the  value  of  the  field  current,  i,  as  function  of  the  time,  t, 
after  closing  the  field  switch. 

After  infinite  time,  t  =  oo ,  the  current  i  assumes  the  final 
value  i'o,  which  is  given  by  substituting  t  =  co  into  equation 
(67),  thus, 

•      e     500     „ 

i0=- =77™  =2  amperes;        •     •     •     •     (()<^) 
r     2oU 

hence,  by  substituting  (68)  into  (67),  this  equation  can  also  be 
written, 

t=to(x-rir() 

=  2(l-£-^'), (69) 

where  io=2  is  the  final  value  assumed  by  the  field  current. 
The  time  t\,  after  which  the  field  current  i  has  reached  90 
per  cent  of  its  final  value  i0,  is  given  by  substituting  i  =  0.9i0 
into  (69),  thus, 

0.9H,=to(l-r£fa), 

and 

c--£-(l=0.1. 

Taking  the  logarithm  of  both  sides, 

r 
-j^hlog  £=-1; 


and 

/.  = 

r  los:  ; 


ti— A- (7°) 


POTENTIAL  SERIES   AND   EXPONENTIAL  FUNCTION.     75 

55.  The  inductance  L  is  calculated  from  the  data  given 
in  the  problem.  Inductance  is  measured  by  the  number  of 
interhnkages  of  the  electric  circuit,  with  the  magnetic  flux 
produced  by  one  absolute  unit  of  current  in  the  circuit;  that 
is,  it  equals  the  product  of  magnetic  flux  and  number  of  turns 
divided  by  the  absolute  current. 

A  current  of  i0=2  amperes  represents  0.2  absolute  units, 
since  the  absolute  unit  of  current  is  10  amperes.  The  number 
of  field  turns  per  pole  is  -4000;  hence,  the  total  number  of  turns 
n  =  4X4Q00  =  16,000.  The  magnetic  flux  at  full  excitation, 
or  i0=0.2  absolute  units  of  current,  is  given  as  $=8Xl06  lines 
of  magnetic  force.     The  inductance  of  the  field  thus  is: 

_     n0     16000X8X106     nAn     1MQ    .     .  .,      aAM 

L  =—  = 77—-      -  =  (540  X 109  absolute  units  =  640.7> , 

the  practical  unit  of  inductance,  or  the  henry  (h)  being  109 
absolute  units. 

Substituting  L  =  G40  r  =  250  and  e=500,  into  equation  (G7) 
and  (70)  gives 

t  =  2(l  -£-°-39'), 

and 

'^250^343  =  5-88  SeC (71) 

Therefore  it  takes  about  6  sec.  before  the  motor  field  has 
reached  90  per  cent  of  its  final  value. 

The  reader  is  advised  to  calculate  and  plot  the  numerical 
values  of  i  from  equation   (71),  for 
t  =  0,  0.1,  0.2,  0.4,  0.63  0.8,  1.0,  1.5,  2.0,  3,  4,  5,  6,  8,  10  sec. 

This  calculation  is  best  made  in  the  form  of  a  table,  thus; 


and, 
hence, 
an<  1 , 


«-0J«=iV  _  o.39*  log  e, 
logs    =0.4343; 

0.39/ log,    -0.1094/; 


e-™*  =  N-Q.umt. 


■('» 


ENGINEERING   MA  Til  EM  A  TI(  'S. 


The   values   of  e    039t  can   also    be    taken    directly   from    the 
tables  of  the  exponential  function,  at  the  end  of  the  book. 


--0.39/ 

/ 

0.1694/ 

-0.1694/ 

=  -V-(j.lb94/ 

l_£—  0.39/ 

2(1  -;-•"•'•" 

0.0 

0 

0 

1 

0 

0 

0.1 

0.0170 

0.9830-1 

0.962 

0.038 

0 .  076 

0.2 

0.0339 

0.9661-1 

0.925 

0 .  075 

0.150 

0.4 

0 . 067S 

0.9322-1 

0.855 

0.145 

0.290 

0.6 

0.1016 

0.8984-1 

0.791 

0.209 

0.418 

0.8 

0.13.55 

0.8645-1 

0 .  732 

0.268 

0 .  536 

etc. 

EXAMPLE    2. 

56.  A  condenser  of  20  inf.  capacity,  is  charged  to  a  potential 
of  e0  =  10,000  volts,  and  then  discharges  through  a  resistance 
of  2  megohms.     What  is  the  equation  of  the  discharge  current, 

and  after  how  long  a  time  has 
the  voltage  at  the  condenser 
dropped  to  0.1  its  initial  value? 
A  condenser  acts  as  a  reser- 
voir of  electric  energy,  similar 
to  a  tank  as  water  reservoir. 
If  in  a  water  tank,  Fig.  27,  A 
is  the  sectional  area  of  the  tank, 
e,  the  height  of  water,  or  water 
pressure,  and  water  flows  out 
of  the  tank,  then  the  height  e 
decreases  by  the  flow  of  water: 
that  is  the  tank  empties,  and 
the  current  of  water,  i,  is  proportional  to  the  change  of  the 

de 

water  level  or  height  oi  water,  — ,  and   to   the   area  A  of   the 


4 A > 

1 

1 

I 

1         V 


Fig.  27.     Water  Reservoir. 


tank;  that  is,  it   is, 


di 


,  de 

i .  =  —  4  — 
di  * 


(72) 


The  minus  sign  stands  on  the  right-hand  side,  as  for  positive 
i;  that  is,  out-flow,  the  height  of  the  water  decreases;  that  is, 
de  is  negative. 


POTENTIAL  SERIES  AND  EXPONENTIAL    FUNCTION.     77 

In  an  electric  reservoir,  the  electric    pressure   <>r  voltage  e 
corresponds  to  the  water  pressure  or  height  of  the  water,  and 

to  the  storage  capacity  or  sectional  area  A  of  the  water  tank 
Corresponds  the  electric  storage  capacity  of  the  condenser, 
called  capacity  C.  The  current  or  flow  out  of  an  electric 
condenser,  thus  is, 

•— cs <™ 

The  capacity  of  condenser  is, 

C  =  20  mf  =  20  X 10-  6  farads. 

The  resistance  of  the  discharge  path  is, 

r  =  2Xl06  ohms; 

hence,  the  current  taken  by  the  resistance,  r,  is 

.     e 
r 

and  thus 

de     e 


—  C 

J  dt     r 


and 

de 


dt  =      Cr  e' 


Therefore,  from  (CO)  (61), 


< 


e  =  a0£   Cr, 
and  for  t  =  0,  e  =  e0=  10,000  volts;  hence 

10,000  =  a0, 

and 

e-  e0e    & 

=  10,000£-°025f  volts;  .     .     .     (74) 
0.1  of  the  initial  value: 

e=0.1e0, 
is  reached  at . 

U=r^—  =  02  sec (75) 

log  £ 


78  ENGINEERING    MA  THEMATIC 'S. 

The  reader  is  advised  to  calculate  and  plot  the  numerical 

values  of  e,  from  equation  (7-1),  for 

2  =  0;  2;  4:  6;  8;   10:  15;  20;  30;   10;  00;  80;  100;  150;  200 sec. 

57.  Wherever  in  an  electric  circuit,  in  addition  to  resistance, 

inductance   and   capacity    both   occur,    the    relations    between 
currents  and  voltages  lead  to  an  equation  containing  the  second 
differential  coefficient,  as  discussed  above. 
The  simplest  form  of  such  equation  is: 

d*  =  ay (/G) 

To  integrate  this  by  the  method  of  indeterminate  coefficients, 
we  assume  as  solution  of  the  equation  (76)  the  infinite  series, 

y  =  a0+aiX  +  a2x2+a'iX3jra^x4  + (77) 

in  which  the  coefficients  a0,  oi,  02,  a3,  aA.  .  .  are  indeterminate. 
Differentiating  (77)  twice,  gives 

-ri=2a2 -f  2  X Sa3x  +  3 X  ia*x2  +  4  X  oa5x3  + .  .  .  ,  .  (78) 
dx2 

and  substituting  (77)  and  (78)  into  (76)  gives  the  identity, 

2a2  +  2  X  Sa3x  +  3  X  4a4x2  +  4  X  5a5.r3  + .  .  . 

=a(a,o+a\X+a2X2  +  a3x3  +  .  .  .); 

or,  arranged  in  order  of  x, 

(2a2-  aa{))  +  x(2X3a3-  acii)  +x2(3x4a4-  acio) 

+  x3(4x5a5-aa3)+.  .  .=0 (79) 

Since  this  equation  (79)  is  an  identity,  the  coefficients  of 
all  powers  of  x  must  individually  equal  zero.  This  gives  for 
the  determination  of  these  hitherto  indeterminate  coefficients 
the  equations, 

2a,2— aao=0; 
2x3a3-aai=0; 

3x4a4  —  aao  =  0; 
4  x5a-,—  aa3=0,  etc. 


POTENTIAL  SERIES  AND  EXPONENTIAL   FUNCTION.     79 


Therefore 

aa() 

<*2=— ; 

aci\ 

aa2      a0a2 

n  ■  —  —          — 

as~rxT- 

a\a2 

a*     3X4"    |4    ' 

|5   ' 

aa,i     a0a3 

a6=5^<Tr"j(r; 

«ia3 

~fT; 

aat;      a0a4 

aa7 

fin  = = 

aia4 

°     7X8      |8   '  y    8X9      |9   ' 

etc.,  etc. 

Substituting  these  values  in  (77), 


ax2     a2x4     a3x6 


</=«„|i+1-+-|j-+-g-+...j 

or3     a2i--5     a3x7 


+ailxV+y+y+---j-  (80) 

In  this  case,  two  coefficients  oo  and  ai  thus  remain  inde- 
terminate, as  was  to  be  expected,  as  a  differential  equation 
of  second  order  must  have  two  integration  constants  in  its 
most  general  form  of  solution. 

Substituting  into  this  equation, 

b2  =  a; 
that  is, 

b  =  Va,       (81) 

cl2v 
and 


f        b2x2     64.r4     lArG  1 

2/=ao{i+^+^r+-j(r+--.} 


a,  L        b3x3     b5xr>     Wx7 
+-±UX+        +        +        +...    .     (83) 


80  ENGINEERING  MATHEMATICS. 

In  this  case,  instead  of  the  integration  constants  ao  and  a\, 
the  two  new  integration  constants  A  and  B  can  be  introduced 
by  the  equations 

a0  =  A+B    and     °^  =  A-B; 

hence>  ai  fli 

a°  +  T  a°~~b 

A=—^—    and     B  =  —^ — > 

and,  substituting  these  into  equation  (83),  gives, 

f        7       62x2    63x3     64x4  1 

2/  =  il  J  l+kr  +  ^+-— +-nj-  +  ...  J 

j,  r  &2^2  ^3  ^4      l 


The  first    series,  however,  from  (56),  for  n  =  &  is  e+bx,  and 
the  second  series  from  (56),  for  n=  —  b  is  e~~hx. 
Therefore,  the  infinite  series  (83)  is, 

y  =  Ae  +  bx+Be-bx; (85) 

that  is,  it  is  the  sum  of  two  exponential  functions,  the  one  with 
a  positive,  the  other  with  a  negative  exponent. 
Hence,  the  differential  equation, 

d2y  nas 

is  integrated  by  the  function, 

y==Ae+hx+Bs-bx, (86) 

where, 

b  =  Va. (87) 

However,  if  a  is  a  negative  quantity,  6  =  Va  is  imaginary, 
and  can  be  represented  by 

b=jc, (88) 

where 

c2=  -a (89) 

In  this  case,  equation  (86)  assumes  the  form, 

y  =  Ae+icx+Be-'cx; (90) 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     81 

that  is,  if  in  the  differential  equation  (76)  a  is  a  positive  quantity, 

=  +  b2,  this  differential  equation  is  integrated  by  the  sum  of 
the  two  exponential  functions  (86);  if,  however,  a  is  a  negative 
quantity,  =  —  c2,  the  solution  (86)  appears  in  the  form  of  exponen- 
tial functions  with  imaginary  exponents  (90). 

58.  In  the  latter  ease,  a  form  of  the  solution  of  differential 
equation  (76)  can  be  derived  which  does  not  contain  the 
imaginary  appearance,  by  turning  back  to  equation  (80),  and 
substituting  therein  a=  —  c2,  which  gives, 


fy 

dx2 


c2y 


(91) 


y=a0\  1 


c2.r2      cV4     (fix? 


2 


li        E 


+  , 


or,  writing  A  =a0  and  B  = 


«i 


c 


r 


y=An 


\  ex 


1 


C2X2        C4X4        C6X6 

16  J 


c3.r3     c5.r5 


c3x3      c5x5 


o 


+  . 


+  . 


(92) 


The  solution  then  is  given  by  the  sum  of  two  infinite  series, 
thus, 


and 


as 


u (ex)  =  1 


v(cx)=cx- 


c2x2      c\v+     c6x6 


<) 


+ 


f3r3         ,..-,,.-. 


10 


(93) 


y=Au(cx)+Bv(cx) (94) 


In  the  w-series,  a  change  of  the  sign  of  x  does  not  change 
the  value  of  u, 

u(  —  cx)  =  u(+cx) (95) 


Such  a  function  is  called  an  even  function. 


82  ENGINEERING  MATHEMATICS. 

In  the  u-series,  a  change  of  the  sign  of  x  reverses  the  sign 
of  v,  as  seen  from  (93) : 

v(—cx)  =  —v(+cx) (96) 

Such  a  function  is  called  an  odd  function. 
It  can  be  shown  that 

u(cx)  =cos  ex    and    v(cx)=  sin  c.r;    .     .     .     (97) 

hence, 

y= A  cos  ex  +B  sin  ex, (98) 

where  A  and  B  are  the  integration  constants,  which  arc  to  be 
determined  by  the  terminal  conditions  of  the  physical  problem. 
Therefore,  the  solution  of  the  differential  equation 

(l2y  men 

d?=av>      f99) 

has  two  different  forms,   an   exponential  and  a  trigonometric. 
If  a  is  positive 

%=+h2v, noo) 

it  is: 

y  =  Ae+bx+Bs-bx,         (101) 

If  a  is  negative, 


d2  n 

U=~c2y> (102) 

it  is: 

y  =  A  cos  ex  +  B  sin  c.r (103) 

In  the  latter  case,  the  solution  (101)  would  appear  as   ex- 
ponential function  with  imaginary  exponents; 

y  =  Ae+icx+Be-icx        (104) 

As  (104)  obviously  must  be  the  same  function  as  (103),  it 
follows  that  exponential  functions  with  imaginary  exponents 
must  be  expressible  by  trigonometric  functions. 


POTENTIAL  SERIES  AND  EXPONENTIAL   FUNCTION.     83 


59.  The  exponential  functions  and  the  trigonometric  func- 
tions, according  to  the  preceding  discussion,  are  expressed  by 
the  infinite  series, 


x2     xs     .r4     x5 


£*  =  l+£  +  —  +nj-  +rr  +tv 
2       \6  5 


.r2     .r4     .r6 


cos^l-^  +  u-^ 


x3     .r5    x7 
sin  x  =  X— irr +iv  —  t=-  H — .  .  . 

•">      5      1/ 


.     (105) 


Therefore,  substituting  ju  for  x, 


CJU. 


IT 


>/::     H4 


It5      ;^G 


1+7^- 2--/T3+T4 +^|5 -re -7|7  " 


u2     u4     II6 
2  o 


IT       //•'       //' 


+  /    W-nr+rr- 


5      ,, 


However,  the  first  part  of  this  series  is  cos  w,  the   latter  part 
sin  u,  by  (105);    that  is, 


eJU  =  cos  u  +j  sin  u. 

Substituting  —u  for  -I  ;/  gives, 

s~>u=  cos  u  —j  sin  //. 
Combining  (106)  and  (107)  gives, 


COS  M  = 


and 


sin  m 


r  +  JIt £~}" 


Substituting  in  ( L06)  to  (108),  jv  for  u,  gives, 

£_v  =  cos   jv+j  sin   ,/> 


and, 


.  -;-  v 


cos    jv—j  sin    jv.  J 


106) 


(107 


(10S) 


(109) 


84  ENGINEERING    MA  Til  EM  A  TICS. 

Adding  and  subtracting  gives  respectively, 

cos  p  = , 

and  f         .     .     .     .     (110) 

sin  iv  = ^- — . 

By  these  equations,  (100)  to  (110),  exponential  functions 
with  imaginary  exponents  can  be  transformed  into  trigono- 
metric functions  with  real  angles,  and  exponential  functions 
with  real  exponents  into  trignometric  functions  with  imaginary 
angles,  and  inversely. 

Mathematically,  the  trigonometric  functions  thus  do  not 
constitute  a  separate  class  of  functions,  but  may  be  considered 
as  exponential  functions  with  imaginary  angles,  and  it  can  be 
said  broadly  that  the  solution  of  the  above  differential  equa- 
tions is  given  by  the  exponential  function,  but  that  in  this 
function  the  exponent  may  be  real,  or  may  be  imaginary,  and 
in  the  latter  case,  the  expression  is  put  into  real  form  by  intro- 
ducing the  trigonometric  functions. 

EXAMPLE    1. 

6o.  A  condenser  (as  an  underground  high-potential  cable) 
of  20  mf.  capacity,  and  of  a  voltage  of  e0  =  10,000,  discharges 
through  an  inductance  of  50  mh.  and  of  negligible  resistance, 
What  is  the  equation  of  the  discharge  current? 

The  current  consumed    by  a  condenser  of  capacity  C  and 

potential  difference  e  is    proportional  to  the  rate  of  change 

of  the  potential  difference,  and  to  the  capacity;    hence,  it    is 

de 
C  — ,  and  the  current   from  the  condenser;    or    its  discharge 

current,  is 

„de 

l=-cit <U1> 

The  voltage  consumed  by  an  inductance  L  is  proportional 
to  the  rate  of  change  of  the  current  in  the  inductance,  and  to  the 
inductance;    hence, 

,  di 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     85 


Differentiating  (112)  gives, 

de       d2i 
dT  'di?' 

and  substituting  this  into  (111)  gives, 

•        rTdH             dH 

1  . 

~CLl 

(113) 
as  the  differential  equation  of  the  problem. 

This  equation  ( 1 1 3 >  is  the  same  as  (102),  for  c2  =  jyj,  thus 
is  solved  by  the  expression, 

^Am*vm+Bsin^u;-   ■  ■  ■  (114) 

and  the  potential  difference  at  the  condenser  or  at  the  inductance 
is,  by  substituting  (114)  into  (112), 

e  =  -\ ITT  \  B  cos — =■  —  A  sin — =•[.      .     (115) 
\C  {  VLC  VLCl 

These  equations  (114)  and  (115)  still  contain  two  unknown 
constants,  A  and  B,  which  have  to  be  determined  by  the  terminal 
conditions,  that  is,  by  the  known  conditions  of  current  and 
voltage  at  some  particular  time. 

At  the  moment  of  starting  the  discharge:  or,  at  the  time 
/  =  0,  the  current  is  zero,  and  the  voltage  is  that  to  which  the 
condenser  is  charged,  that  is,  i=0,  and  e  =  e0. 

Substituting  these  values  in  equations  (114)  and  (115) 
gives, 

[L 


hence 


0  =  A     and     e0=^\jrB; 


B=e0JI, 


and,  substituting  for  A  and  B  the  values  in  (114)  and   (115), 


gives 


(■     .         t 


i  =  en-\  rr  sin 


°\£        VCL' 

and 

t 
e  =  en  cos  — -= 
0        VCL 


(110) 


86  ENGINEERING  MATHEMATH  'S. 

Substituting  the  numerical  values,  e0  =  10,000  volts,  C= 20 
mf.  =  20 X 10- 6  farads,  L=50  mh.=0.05h.  gives, 

fy  =  0.02    and     \  CL  =  10"3; 

hence, 

i  =  2()()  sin  1000  /    and     e= 10,000  cos  1000  t. 

6i.  The  discharge  thus  is  alternating.  In  reality,  due  to 
the  unavoidable  resistance  in  the  discharge  path,  the  alterna- 
tions gradually  die  out,  that  is,  the  discharge  is  oscillating. 

The  time  of  one  complete  period  is  given  by, 

2- 
1000f0  =2tt;     or,     *o  =  -[000- 


Hence  the  frenquency, 
1 

7 


/=  —  =-?; —  =  159  cvcles  per  second,, 
'o       -" 


As  the  circuit  in  addition  to  the  inductance  necessarily 
contains  resistance  r,  besides  the  voltage  consumed  by  the 
inductance  by  equation  (112),  voltage  is  consumed  by  the 
resistance,  thus 

er  =  ri, (117) 

and  the  total  voltage  consumed  by  resistance  r  and  inductance 
L,  thus  is 

•     r  di 
e=n+Ljt (118) 

Differentiating  (118)  gives. 

de       di     T  d-i  ,„„   ^ 

irrdt+Lw    <"'■» 

and,  substituting  this  into  ('([nation  (111),  gives, 

,   di         ,  d-i     „ 

t+Crdi+CLW*=0' (120) 

as  the  differential  equation  of  the  problem. 

This  differentia]  equation  is  of  the  more  general  form,  (30), 
62.  The  more  general  differential  equation  (30). 

cPy        du 

^J+2cfx+ay+b-0 (121) 


POTENTIAL  SERIES  AND  EXPONENTIAL   FUNCTION.     87 
can,  by  substituting, 


//I  Z  •  •  •  •  •  •  •  •  «  (       llM^     J 

a 


which  (rives 


&' 


dy    dz 
dx    dx' 

bo  transformed  into  the  somewhat  simpler  form, 

d%        dz  ,_    lS 

It  may  also  be  solved  by  the  method  of  indeterminate 
coefficients,  by  substituting  for  z  an  infinite  series  of  powers  of 
x,  and  determining  thereby  the  coefficients  of  the  series. 

As,  however,  the  simpler  forms  of  this  equation  were  solved 
by  exponential  functions,  the  applicability  of  the  exponential 
functions  to  this  equation  (123)  may  be  directly  tried,  by  the 
method  of  indeterminate  coefficients.  That  is,  assume  as  solu- 
tion an  exponential  function, 

z  =  Aebx, (124) 

where   A   and   b  are   unknown   constants.     Substituting   (124) 
into  (123),  if  such  values  of  A  and  6  can  be  found,  which  make 
the  substitution   product   an   identity,    (124)   is  a  solution  of 
the  differential  equation  (123). 
From  (124)  it  follows  that, 

~  =  bAebx;     and     ~=b^A£-bx      .     .     (125^ 
dx  dx2 

and  substituting  (124)  and  (125)  into  (123),  gives, 

Aebx{b2+2cb+a\=0 (126) 

As  seen,  this  equation  is  satisfied  for  every  value  of  x,  that 
is,  it  is  an  identity,  if  the  parenthesis  is  zero,  thus, 

b2+2cb+a  =  0 (127) 

and  the  value  of  h,  calculated  by  the  quadratic  equation  (127). 
thus  makes  (124)  a  solution  of  (123),  and  leaves  -I  still  undeter- 
mined, as  integration  constant. 


88  ENGINEERING   MATHEMATICS. 

From  (127), 

b=—c±\//c2—a; 
or,  substituting, 

Vc2-a  =  p, (128) 

into  (128),  the  equation  becomes, 

b=-c±p (129) 

Hence,  two  values  of  b  exist, 

&i=  —  c  +  p      and      62=—  c— p, 

and,  therefore,  the  differential  equation, 

d~z         dz 

_+2^+a*=0, (130) 

is  solved  by   AebiX;   or,   by   An1'™ ,  or,   by  any  combination  of 
these  two  solutions.     The  most  general  solution  is, 

z  =  A1ebiX+A2ebiX; 
that  is, 

1} 


y  =  Ai£(-c  +  p)x+A2S{-c-p)x- 

=  e~cx\Ai£+px  +  A2e~px\  — 


ia\.      ■     ■     (131) 

b 


a 


As  roots  of  a  quadratic  equation,  61  and  b2  may  both  be 
real  quantities,  or  may  be  complex  imaginary,  and  in  the 
latter  case,  the  solution  (131)  appears  in  imaginary  form,  and 
has  to  be  reduced  or  modified  for  use,  so  as  to  eliminate  the 
imaginary  appearance,  by  the  relations  (100)  and  (107). 

EXAMPLE  2. 

63.  Assume,  in  the  example  in  paragraph  60,  the  discharge 

circuit   of   the   condenser   of   C  =  20   inf.    capacity,    to   contain, 

besides  the  inductance,  L  =  0.05  h,  the  resistance,  r  =  125  ohms. 

The  general  equation  of  the  problem,   (120),  dividing  by 

C  />,  becomes, 

d2i     r  di      i  , 

df>+Ldt+CL=0 (132> 


POTENTIAL  SERIES  AND  EXPONENTIAL   FUNCTION.     89 


This  is  the  equation  (123),  for 


x  =  t,    2c=-£  =  2500; 


If 


and,  writing 


z 

=h 

p= 

a  = 

1 
~CL 

=  106 

\    r- 

-a, 

then 

rV 

1 

p= 

2L\ 

s)" 

~CL 
L 

/,-< 

i- 

4L 

s 

V 

=  2Z' 

and  since 


7rr  =  10     and     77=2500,  | 


s  =  75     and      2)=~")()- 
The  equation  of  the  current  from  (131)  then  is, 


i=Axl      2L       +  X2£      2L 


r  s  .« 


J  •  J 


.     (133) 


(134) 


(135) 


(136) 


(137) 


This  equation  still  contains  two  unknown  quantities,  the  inte- 
gration constants  A\  and  A2,  which  are  determined  by  the 
terminal  condition:  The  values  of  current  and  of  voltage  at  the 
beginning  of  the  discharge,  or  t  =  0. 

This  requires  the  determination  of  the  equation  of  the 
voltage  at  the  condenser  terminals.  This  obviously  is  t he  voltage 
consumed  by  resistance  and  inductance,  and  is  expressed  by 
equation  (118), 


e=n+L-g, 


(118) 


90 


ENGINEERING   MA  THEMATICS. 


hence,  substituting  herein  the  value  of  i  and  -r,  from  equation 
(137),  gives 


~*t 


\-L\    ^rAl£   2L*-.-Z-A2e    u  | 


J        i 


\    -iL 


2L 


r  +  sA     -L^ft     r-sA     -r-±*t 
Ait     1L    +—^-A2z     2L 


■if* 


±-t 


rJlAl£+*Ur-^A2s 


ii:jN.) 


and,  substituting  the  numerical  values    (133)   and    (130)   into 
c« illations    (137)   and  (138),  gives 


(100/ 


and, 


i  =  A1e-500t+A2e-2 


e=lOOA1e-500t+25A2£ 


139) 


2000/ 


J 


At    the   moment  of  the   beginning  of  the   discharge,   t=0, 

the  current  is  zero  and  the  voltage  is  10,000;   that  is, 


.(140) 


/=();  /  =  0;  e  =  10,000   .... 

Substituting  (140)  into  (139)  gives. 

0  =  A1+A2,     10,000  =  100Ai+25A2; 
hence, 

A2=   -AX]     A!=133.3;     .1.         133.3.    .     .     (141) 

Therefore,  the  current  and  voltage  arc, 


;=133.3{£-500t-e-2000'i, 
c=13,333s-5O0(-3333£    2000< 


(142) 


The  reader  is  advised  to  calculate  and   plot    the  numerical 
values  of  i  and  e,  and  of  their  two  components,  for, 


1  =  0,  0.2,  0.4,  0.6,  1.  L.2,  L.5,  2,  2.5,  3,  1.  5,  6  XlO-3  sec. 


POTENTIAL  SERIES  AND  EXPONENTIAL   FUNCTION.     91 

64.  Assuming,  however,  that  the  resistance  of  the  discharge 

circuit  is  only  r  =  80  ohms  (instead  of   L25  ohms,  as  assumed 
above) : 

4L. 

r2—  y?  m  equation  (134)  then  becomes  3600,  and  there- 
fore : 

s  =  V-3600  =  60  y/^l  =  ( ;( )jf 

and 

P  =  2X  =  M0j. 

The  equation  of  the  current  (137)  thus  appears  in  imaginary 
form, 

i  =  £-800'{  A1e+600it  +  A2e-600it}.       .     .     .     (143) 

The  same  is  also  true  of  the  equation  of  voltage. 

As  it  is  obvious,  however,  physically,  that  a  real  current 
must  be  coexistent  with  a  real  e.m.f.,  it  follows  that  this 
imaginary  form  of  the  expression  of  current  and  voltage  is  only 
apparent,  and  that  in  reality,  by  substituting  for  the  exponential 
functions  with  imaginary  exponents  their  trigononetric  expres- 
sions, the  imaginary  terms  must  eliminate,  and  the  equation 
(143)  appear  in  real  form. 

According  to  equations  (106)  and  (107), 

£+6003*= cos  600*+/  sin  (100/ : 

e-Q00jt=cos  QOOt-j  sm  ('»()()/. 

Substituting  (144)  into  (143)  gives, 

i  =  s  -  800/  J  B  i  cos  000/  +  B2  sin  600/ } ,      .     .     (145) 

where  Bi  and  B2  are  combinations  of  the  previous  integration 
constants  A\  and  A2  thus, 

Bi  =  Al+A2,    and     B2=j{Ax-A2).   .     .      (146) 

By  substituting  the  numerical  values,  the  condenser  e.m.f., 
given  by  equation  (138),  then  becomes, 

e=£-soon  (40+30j).4i(cos  000/  +j  sin  000/) 

+  (40- 30 j) A 2(cos  000/ -/sin  000/)  | 
=  e~800')  (40Bi  +3052)cos  000/  +  (40S2-30£i)  sin  000/ 1.     (147) 


(144) 


92  ENGINEERING  MATHEMATICS. 

Since  for  t=0,  i  =  0  and  e  =  10,000  volts  (140),  substituting 

into  (145)  and  (147), 

0=J?i  and  10,000  =  40  Bi+30  B2. 
Therefore,  Bi=0  and  B2=333  and,  by  (145)  and  (147), 
?:=333£-800'sin  600  t; 
e=l(),00()£-800'  (cos  600  £  +  1.33  sin  600 i). 


(IIS) 


As  seen,  in  this  case  the  current  i  is  larger,  and  current 
and  e.in.f.  are  the  product  of  an  exponential  term  (gradually 
decreasing  value)  and  a  trigonometric  term  (alternating  value) ; 
that  is,  they  consist  of  successive  alternations  of  gradually 
decreasing  amplitude.  Such  functions  are  called  oscillcdhuj 
functions.  Practically  all  disturbances  in  electric  circuits 
consist  of  such  oscillating  currents  and  voltages. 

600/  =  2x  gives,  as  the  time  of  one  complete  period, 

2r 


T  =  —-  =  0.0105  sec; 
600 


and  the  frequency  is 


/=  777  =  95.3  cycles  per  sec. 

In  this  particular  case,  as  the  resistance  is  relatively  high, 
the  oscillations  die  out  rather  rapidly. 

The  reader  is  advised  to  calculate  and  plot  the  numerical 

values  of  i  and  e,  and  of  their  exponential  terms,  for  every  30 

T       T        T 
degrees,  that  is,  for  t  =  0,  zr»,  2  y^,  3^,  etc.,  for  the  first  two 

periods,  and  also  to  derive  the  equations,  and  calculate  and  plot 
the  numerical  values,  for  the  same  capacity,  C  =  20  inf.,  and 
same  inductance,  L  =  0.()5//,  but  for  the  much  lower  resistance, 
r  =  20  ohms. 

65.  Tables  of  e+x  and  e~x,  for  5  decimals,  and  tables  of 
log  s+x  and  log  e~x,  for  (i  decimals,  are  given  at  the  end  of 
the  book,  and  also  a  table  of  s~x  for  3  decimals.  For  most 
engineering  purposes  the  latter  is  sufficient ;  where  a  higher 
accuracy  is  required,  the  5  decimal  table  may  be  used,  and  for 


POTENTIAL  SERIES  AND  EXPONENTIAL  FUNCTION.     93 

highest  accuracy  interpolation  by  the  logarithmic  table  may  be 

employed.     For  instance, 

e—  13.6847  _? 

From  the  logarithmic  table, 

log  c-io  =5.057055, 
log  £-3  =8.097117, 
log  e-o-6  =9.739423, 
log  £-o.o8  =  9.965256, 

r  interpolated, 
log  £-0.0047  =  9.997959  I  between  log  e-0-004  =9.998203, 

[  and  logs"0005  =9.997829), 

added 

log  £-13-6847 = 4.056810 =0.056810  -6. 

From  common  logarithmic  tables, 

£-13.6847  =  113976xl0-6> 

Note.  In  mathematics,  for  the  base  of  the  natural  loga- 
rithms, 2.718282  .  .  .  ,  is  usually  chosen  the  symbol  e.  Since, 
however,  in  engineering  the  symbol  e  is  universally  used  to 
represent  voltage,  for  the  base  of  natural  logarithms  has  been 
chosen  the  symbol  e,  as  the  Greek  letter  corresponding  to  e, 
and  e  is  generally  used  in  electrical  engineering  calculations  in 
this  meaning. 


CHAPTER  III. 
TRIGONOMETRIC  SERIES. 

A.   TRIGONOMETRIC    FUNCTIONS. 

66.  For  the  engineer,  and  espeeially  the  electrical  engineer. 
a  perfect  familiarity  with  the  trigonometric  functions  and 
trigonometric  formulas  is  almost  as  essential  as  familiarity  with 
the  multiplication  table.  To  use  trigonometric  methods 
efficiently,  it  is  not  sufficient  to  understand  trigonometric 
formulas  enough  to  be  able  to  look  them  up  when  required, 
but  they  must  be  learned  by  heart,  and  in  both  directions;  that 
is,  an  expression  similar  to  the  left  side  of  a  trigonometric  for- 
mula must  immediately  recall  the  right  side,  and  an  expression 
similar  to  the  right  side  must  immediately  recall  the  left  side 
of  the  formula. 

Trigonometric  functions  are  defined  on  the  circle,  and  on 
the  right  triangle. 

Let  in  the  circle,  Fig.  28,  the  direction  to  the  right  and 
upward  be  considered  as  positive,  to  the  left  and  downward  as 
negative,  and  the  angle  a  be  counted  from  the  positive  hori- 
zontal OA,  counterclockwise  as  positive,  clockwise  as  negative. 

The  projector  s  of  the  angle  a,  divided  by  the  radius,  is 
called  sin  a ;  the  projection  c  of  the  angle  a,  divided  by  the 
radius,  is  called  cos  a. 

The  intercept  /  on  the  vertical  tangent  at  the  origin  .1. 
divided  by  the  radius,  is  called  tan  a;  the  intercept  ct  on  the 
horizontal  tangent  at  B,  or  90  deg.,  behind  A,  divided  by  the 
radius,  is  called  cot  a. 


Thus,  in  Fig.  28, 


s  c 

sina=-;    cosa  =  -; 
r  r 


t  el 

tan  «  =  -;     cot  a  =  — . 
r  r 


(1 


94 


TRIGONOMETRIC  SERIES. 


95 


In   the  right  triangle,   Fig.  29,  with  the  angles  a  and  /?, 
opposite  respectively  to  the  cathetes  a  and  b,  and  with  the 

hypotenuse  c,  the  trigonometric  functions  are: 


sm  o'  =  cos  /?  =  -;     cos  a=sin /?  =  - 
f      c  c 


.      •     •     (2) 


tan  a  =  cot . /?  =  y  ',    cot  a=tan /?  =  — . 


By  the  right  triangle,  only  functions  of  angles  up  to  90  (leg., 

or    -,  can  be  defined,  while   by  the  circle  the  trigonometric 

functions  of  any  angle  are  given.     Both  representations  thus 
must  be  so  familiar  to  the  engineer  that  he  can  see  the  trigo- 


Fig.  28.     Circular  Trigonometric 

Functions. 


Fir;.  29.     Triangular  Trigono- 
metric Functions. 


nometric  functions  and  their  variations  with  a  change  of  the 
angle,  and  in  most  cases  their  numerical  values,  from  the 
mental  picture  of  the  diagram. 

67.  Signs  of  Functions.  In  the  first  quadrant,  Fig.  28,  all 
trigonometric  functions  are  positive. 

In  the  second  quadrant,  Fig.  30,  the  sin  a  is  still  positive, 
as  s  is  in  the  upward  direction,  but  cos  a  is  negative,  since  <■ 
is  toward  the  left,  and  tan  a  and  cot  a  also  are  negative,  as  / 
is  downward,  and  ci  toward  the  left. 

In  the  third  quadrant,   Fig.  31,  sin  a  and  cos  a  are  both 


% 


EN(  SNEERING   M.  1  THEM  A  TI(  'S. 


negative:   s  being  downward,  c  toward  the  left;   but  tan  a  and 

cot  a  arc  again  positive,  as  seen  from  t  and  ct  in  Fig.  31. 


Fig.  30.     Second  Quadrant. 


Fig.  31.    Third  Quadrant. 


In  the  fourth  quadrant,  Fig.  32,  sin  a  is  negative,  as  s  is 
downward,  but  cos  a  is  again  positive,  as  c  is  toward  the  right; 

tan  a  and  cot  a  are  both 
negative,  as  seen  from  t  and 
ct  in  Fig.  32. 

In  the  fifth  quadrant  all 
the  trigonometric  functions 
again  have  the  same  values 
as  in  the  first  quadrant,  Fig. 
28,  that  is,  360  deg.,  or  2-, 
or  a  multiple  thereof,  can  be 
added  to,  or  subtracted  from 
the  angle  a,  without  changing 
the  trigonometric  functions, 
but  these  functions  repeat 
after  every  360  deg.,  or  2-- 


Fig.  32.     Fourth  Quadrant. 


thai  i :,  have  2-  or  300  deg.  as  their  perio< 

SIGNS   OF   FUNCTIONS 


Function. 

Positive. 

Negative. 

sin  a 
cos  a 
tan  (t 
col  a 

1st  and  2d 
1st  and   llh 
Isl  and  3d 
Lsl  and  3d 

3d    and  Ith  quadrant 
2d    and  3d 
2d    and  Ith 

2d    and    llh 

(3) 


TRIGONOMETRIC  SERIES.  97 

68.  Relations  between  sin  a  and  cos  a.  Between  sin  a  and 
cos  a  the  relation, 

sin2a+cos2a  =  l, (4) 

exists;  hence, 

sin  a  =  Vl-cos2  a;  }  „. 

cos  «  =  v/l  — sin2  «.   J 

Equation  (4)  is  one  of  those  which  is  frequently  used  in 
both  directions.  For  instance,  1  may  be  substituted  for  the 
sum  of  the  squares  of  sin  a  and  cos  a,  while  in  other  cases 
sin2  a  +cos2  a  may  be  substituted  for  1.     For  instance, 

sin2  a  +  cos2  a 


cos2  a  cos2  a 


/sm«\-     .  0 

I +l=tan2a 

\cos  a/ 


Relations  between  Sines  and  Tangents. 

sin  a     ) 
tan  a  = ; 

C0S"'i (5) 

cos  a 
cot  a=-^-   -; 
sin  a     J 

hence 

1      .1 


cot  a  = 


tan  « 


tan  a ' 
1 


\ (5«) 


cot  a'  J 


As  tan  a  and  cot  a  are  far  less  convenient  for  trigonometric 
calculations  than  sin  a  and  cos  a,  and  therefore  are  less  fre- 
quently applied  in  trigonometric  calculations,  it  is  not  neces- 
sary to  memorize  the  trigonometric  formulas  pertaining  to 
tan  a  and  cot  a,  but  where  these  functions  occur,  sin  a  and 
and  cos  a'  are  substituted  for  them  by  equations  (5),  and  the 
calculations  carried  out  with  the  latter  functions,  and  tan  a 
or  cot  a  resubstituted  in  the  final  result,  if  the  latter  contains 

sin  a  . 

,  or  its  reciprocal. 

cos  a 

In  electrical  engineering  tan  a  or  cot  a   frequently  appears 

as  the  starting-point  of  calculation  of   the  phase  of  alternating 

currents.     For  instance,  if  a  is  the  phase    angle  of  a  vector 


'.is 


ENGINEERING   MATHEMATIl  '8. 


quantity,  tan  a  is  given  as  the  ratio  of  the  vertical  component 
over  the  horizontal  component,  or  of  the  reactive   component 
over  the  power  component. 
In  this  case,  if 

a 


tan  a  = 


b' 


a 


sin  a  = 


V«2  +  6 


o> 


and 


cos  a  = 


Va2+b2' 


(5b) 


or,  if 


cot  a=-j. 

a' 


d 


sin  rv  = 


Vc2  +  d-' 


and 


cos  a  = 


Vc2+d2 


(5c) 


The  secant  functions,  and  versed  sine  functions  are  so 
little  used  in  engineering,  that  they  are  of  interest  only  as 
curiosities.     They  are  defined  by  the  following  equations: 

1 


sec  a 


cosec  a 


cos  a 

1 


sin  a 

sin  vers  a  =  1  — sin  a, 

cos  vers  a  =  1  —  cos  a. 

69.  Negative  Angles.  From  the  circle  diagram  of  the 
trigonometric  functions  follows,  as  shown  in  Fig.  33,  that  when 
changing  from  a  positive  angle,  that  is,  counterclockwise 
rotation,  to  a  negative  angle,  that  is,  clockwise  rotation,  s,  t, 
and  d  reverse  their  direction,  but  c  remains  the  same;  that  is, 


sin  (  -a)      -sin  a,   ) 
cos  (  —  a)  =  +cos  a, 
tan  (  — «)  =  -tan  a, 
cot  (  —a)  =      cot  a, 


(6) 


cos  a  thus  is  an    "  even  function,"  while  the  three  others  are 
"  odd  functions." 


TRKiOXOMETRK '  SERIES. 


99 


Supplementary    Angles.      From   the  circle   diagram   of  the 
trigonometric  functions  follows,  as  shown  in  Fig.  34,  that  by   , 
changing  from  an  angle  to  its  supplementary  angle,  s  remains 
in  the  same  direction,  but  r.  t,  and  ct  reverse  their  direction, 
and  all  four  quantities  retain  the  same  numerical  values,  thus, 


sin  (-—«)=  +sin  a,  • 

COS  (~—  a)  =  —  cos  a', 

tan  (rz—a)  =  —tan  a, 

I 
cot  (ic—  a)=  —cot  a.    J 


(7) 


\                ct 

B         ct 

/ 

1               ^ 

■a               yS 

t 

A     . 

I             c     o 

\        c 

1- 

t 

Fig.  33.  Functions  of  Negative  Fig.  34.  Functions  of  Supplementary 

Angles.  Angles. 

Complementary    Angles.     Changing  from  an  angle  a  to  its 
complementary  angle  90°  —  a,  or  „-—  a,  as  seen  from   Fig.  35, 

the  signs  remain  the  same,  but  s  and  c,  and  also  t  and  ct  exchange 
their  numerical  values,  thus, 

sin  I  —  —  a  I  =cos  a  , 


s(~  —a)  =  sin  " 

(H 


(3) 


tanl  o  —  a  )  =<'°'  "  • 


100 


ENGINEERING  MA  Til  EM. \  TI(  'S. 


70.  Angle  (<\  ±7t).     Adding,  or  subtracting  it  to  an  angle  a, 

il 

/ 


gives  the  same  numerical  values  of  the  trigonometric  functions 


Fig.  35.     Functions  of  Cornplemen-     Fig.  36.     Functions  of  Angles  Plus 
tary  Angles.  or  Minus  -. 

as  a,  as  seen  in  Fig.  36,  but  the  direction  of  s  and  c  is  reversed, 
while  t  and  ct  remain  in  the  same  direction,  thus, 

sin  («±~)  =  — sin  a,  -j 
cos  (a  ±7r)  =  —cos  a, 
tan  (a±x)=  -f-tan  a, 
cot  (a  ±tz)  =  +cot  a.  J 


(9) 


FiG.  37.  Functions  of  Angles+  -jr.       Fig.  38.  Functions  of  Angles  Minus  ~. 

Angle!  a  ±~  ).     Adding  :),  or  90  deg.  to  an  angle  n,  inter- 
changes the  functions,  6-  and  c,  and  £  and  c/,  and  also  reverses 


TRIGONOMETRIC  SERIES. 


101 


the  direction  of  the  cosine,  tangent,  and  cotangent,  but  leaves 
the  sine  in  the  same  direction,  since  the  sine  is  positive  in  the 
second  quadrant,  as  seen  in  Fig.  o7. 

Subtracting  — ,  or  90  deg.  from  angle  a,  interchanges  the 

functions,  s  and  c,  and  /  and  ct,  and  also  reverses  the  direction, 
except  that  of  the  cosine,  which  remains  in  the  same  direction; 
that  is,  of  the  same  sign,  as  the  cosine  is  positive  in  the  first 
and  fourth  quadrant,  as  seen  in  Fig.  38.      Therefore, 


sin 


cos 


tan 


cot 


sin 


cos 


tan 


cot 


v  +-  I  =  +cos  a, 


it 


a 


sill  a 


- 


a+- 


«+2, 


(X 


2 


7T  ' 


«-- 


cot  a, 


=  —tan  a, 


=  —cos  a, 


=  +sin  a, 


=  —cot  «, 


(10) 


71 


«  — —  I  =  —tan  a. 


(ID 


Numerical   Values.     From  the  circle  diagram,  Fig.  28,  etc., 
follows  the  numerical  values: 


sin     0°  =  0 
sin    30°  =  J 
sin    45°  =  i\    2 
sin    60°  =  £V3 
sin    90°  =  1 
sinl20°=iV'3 
etc. 


cos      0°=1 
cos    30°  =  *\/3 
cos    45°  =  $\   2 
cos    60°= i 
cos    90°  =  0 
cos  120°      -i 
etc. 


tan      0°  =  0 
tan    45°  =  1 
tan    90° =oo 
tan  135°= -1 
etc. 


cot      0°  =  oo 
cot    45°  =  1 
cot    90°  =  0 
cot  135°=- 1 
etc. 


(12) 


L02 


ENGINEERING   MATHEMATIi  'S. 


\ 


(13) 


71.  Relations  between  Two  Angles.     The  following  relations 
are  developed  in  text-books  of  trigonometry: 

>'ni  (a+/3)=sin  a'  cos  /3+COS  a  sin  /?, 

sin  (a  —  /3)=sin  a  cos  /?— cos  a  sin  /?, 

cos  (a  J  /?)     cosa  cos  -5  —  sin  «  sin  /3, 

cos  (a  —  /?)  =COS  a  cos  /3+sin  a:  sin  [i,  . 

Herefrom    follows,  by  combining    these    equations    (13)    in 
pairs: 

cos  a  cos  /5=4{cos  (a  +/?)  +cos  {a  —  ,8)  j , 

sin  a  sin  /3  =  |{cos  (a  —  ,5)  —  cos  (a  +/?}), 

sin  a:  cos/?  =  |{sin  (a +/?) +sin  (a  —  /?)}, 

cos  a-  sin  /?  =  J  { sin  (a  +/?)  —  sin  (a  —  3 ) ! . 

By  substituting  «i  for  («+/?),  and  /?i  for  (a  — 5)  in  these 
equations  (14),  gives  the  equations, 

sm  ai  +  sin  [3i  =     2  sin  — ~ cos  — ~ — 

_  & 

•     o  o   •     ^i-/9!  ai+^i, 

sm  a  1  —  sm  ,ii  =     2  sin  — ~ —  cos  — ~ — 


(14) 


0  o        «i+0i         ai-  /9i, 

cos  a  1  +  cos  /J  1  =      2  cos  ■ — ^ —  cos  — x — 


cos  a  1  —  cos  /?i  =  —  2  sin 


sm 


(15) 


These  three  sets  of  equations  are  the  most  important  trigo- 
nometric formulas.  Their  memorizing  can  be  facilitated  by 
noting  that  cosine  functions  lead  to  products  of  equal  func- 
tions, sine  functions  to  products  of  unequal  functions,  and 
inversely,  products  of  equal  functions  resolve  into  cosine, 
products  of  unequal  functions  into  sine  functions.  Also  cosine 
functions  show  a  reversal  of  the  sign,  thus:  the  cosine  of  a 
sum  is  given  by  a  difference  of  products,  the  cosine  of  a  differ- 
ence by  a  sum,  for  the  reason  that  with  increasing  angle 
the  cosine  function  decreases,  and  the  cosine  of  a  sum  of  angles 
thus  would  be  less  than  the  cosine  of  the  single  angle. 


TRIGONOMETRIC '  SERIES. 


103 


Double  Angles.     From  (13)  follows,  by  substituting  a  for  /3: 


sin  2«  =  2  sin  a  cos  a, 
cos  2a  =  cos2  a  — sin1'  a 
=  '2  cos2  a  —  1, 
=  1-2  sin2  a. 


Herefrom  follow 

.    ,         1  — cos  2a  ,  _         If  cos  2n 

sm~a=-    -jr-  and       cosz«=-  — h— 


(10) 


(16a) 


72.  Differentiation. 


d 


-7-  (sin  z)  =  +  cos  x, 


L 

dx 


(  cos  x)  =  —sin  2 


(17) 


The  sign  of  the  latter  differential  is  negative,  as  with  an 
increase  of  angle  a,  the  cos  a  decreases. 


Integration. 


sin  ada  =  —cos a', 


Jcos 


"I 


\ 


ada  =  +sina. 


Herefrom  follow  the  definite  integrals : 


sin  (a+a)da  =  0 


cos  (a  +  a)da  =  0; 


J 


X 

1      ° 

I        sin  (a  +  a)da  =  2  cos  (c+a); 

Jc  I 

Xc  +  x 
cos  (a  +  a)da  =  —  2  sin  (c  +  a) ; 


(IS) 


(18a) 


.     .     (186) 


101 


EXdlXEERlXa  MATHEMATH  '8. 


£ 

f 


sin  m/a  =  0 ; 


cos  «do:  =  0; 


smad<x=  +1; 


cosac?a'=  +1. 


(18c) 


(18d) 


73.  Binomial.  One  of  the  most  frequent  trigonometric 
operations  in  electrical  engineering  is  the  transformation  of  the 
binomial,  a  cos  a +b  sin  a,  into  a  single  trigonometric  function, 
by  the  substitution,  a  =  c  cos  p  and  6  =  c  sin  79;   hence, 

a  cos  a  +  &  sin  a=c  cos  (a—  p),        .     .     .     (19) 
where 

h 

.     •     (20) 


c  =  Va2  +  62     and     tan  p = — ;     . 


a 


or,  by  the  transformation,  a  =  c  sin  5  and  6  =  c  cos  g, 
a  cos  a  +  b  sin  a  =  c  sin  (a +  5),    .     . 


(21) 


where 


a 


c  =  Va2'+b2     and     tan</  =  r- (22) 


74.  Polyphase  Relations. 


2^  cos  (a+a±- 


2- 


m    a  +a±- 


n   1 

2  mi-'" 


(23) 


n 


0, 


1  J 

where  m  and  n  are  integer  numbers. 

Proof.     The  points  on  the  circle  which   defines   the   trigo- 
nometric  function,    by    Fig.   28,   of  the  angles  (a  +  a±- 


2miny 


n 


TRIGONOMETRK '  SHU  IKS. 


105 


arc  corners  of  a  regular  polygon,  inscribed  in  the  circle  and 
therefore  having  the  center  of  the  circle  as  center  of  gravity. 
For  instance,  for  n  =  7,  m  =  2,  they  are  shown  as  Pi,  Po,  •  •  •  Pi, 
in  Fig.  39.  The  cosines  of  these  angles  arc  the  projections  on 
the  vertical,  the  sines,  the  projections  on  the  horizontal  diameter, 
and  as  the  sum  of  the  projections  of  the  corners  of  any  polygon, 


Fig.  39.     Polyphase  Relations. 


FiG.  40.     Triangle. 


on  any  line  going  through  its  center  of  gravity,  is  zero,  both 
sums  of  equation  (23)  are  zero. 


2m  i-^ 


2_]  c°s  la+a± 

1  ^ 


/       ,      2mix\     n 
cosl^+oi  )  =ry  cos  {a  —  b), 


n  I 

^yj  sin  la+a± 
"S^  sin  (  a  +  a  ± 


\rm£\    .     / 

)  sin  (  a 

n   J        \ 


n 
2mi7z 


n 


n 


2mi7z\     n 
+  o±-- — )=^cos  (a—b), 


I  2)))i-\     n   .     . 

cos(a+o±-      -  I  =—  sin  (a—  0). 


>  (24) 


These  equations  are  proven  by  substituting  for  the  products 
the  single  functions  by  equations  (14),  and  substituting  them 
in  equations  (23). 

75.  Triangle.  If  in  a  triangle  a,  8,  and  j  are  the  angles; 
opposite  respectively  to  the  sides  a,  h,  c,  Fig.  40,  then, 


sin  a :-Hsin  /?~sin  y  =  a  +  b+c,    . 


(25) 


106  ENGINEERING   MA  T HEMATICS. 

a?+b2-c2 


cos  y 


2ab 


or 


(20) 


Area 


c2  =  a2  +  b2  —  2ab  cos  y. 
ab  sin  y 


2 

c-  sin  a  sin  (3 
2  sin  y 


(27) 


B.   TRIGONOMETRIC    SERIES. 

76.  Engineering  phenomena  usually  are  either  constant, 
transient,  or  periodic.  Constant,  for  instance,  is  the  terminal 
voltage  of  a  storage-battery  and  the  current  taken  from  it 
through  a  constant  resistance.  Transient  phenomena  occur 
during  a  change  in  the  condition  of  an  electric  circuit,  as  a 
change  of  load;  or,  disturbances  entering  the  circuit  from  the 
outside  or  originating  in  it,  etc.  Periodic  phenomena  are  the 
alternating  currents  and  voltages,  pulsating  currents  as  those 
produced  by  rectifiers,  the  distribution  of  the  magnetic  flux 
in  the  air-gap  of  a  machine,  or  the  distribution  of  voltage 
around  the  commutator  of  the  direct-current  machine,  the 
motion  of  the  piston  in  the  steam-engine  cylinder,  the  variation 
of  the  mean  daily  temperature  with  the  seasons  of  the  year,  etc. 

The  characteristic  of  a  periodic  function,  y=f(x),  is,  that 
at  constant  intervals  of  the  independent  variable  x,  called 
cycles  or  periods,  the  same  values  of  the  dependent  variable  y 
occur. 

.Most  periodic  functions  of  engineering  are  functions  of  time 
or  of  space,  and  as  such  have  the  characteristic  of  univalence; 
that  is,  to  any  value  of  the  independent  variable  x  can  corre- 
spond only  one  value  of  the  dependent  variable  //.  In  other 
words,  at  any  given  time  and  given  point  of  space,  any  physical 
phenomenon  can  have  one  numerical' value  only,  and  therefore 
must  be  represented  by  a  univalent  function  of  time  and  space. 

Any  univalent  periodic  function, 

y=f(x) (0 


TRIGONOMETRIC  SERIES. 


10; 


can  be  expressed  by  an  infinite  trigonometric  series,  or  Fourier 
series,  of  the  form, 


y  =  fl0  +  ft  1  cos  ex  +  a2  cos  lex  4-  a3  cos  3c.r  + .  .  .  . 
+  h]  sin  ex  +  b2  sin  2ex  +  &3  sin  Sex  + .  .  .    : 

or,  substituting  for  convenience,  ex  =  0,  this  gives 

y  =  fto  +  ft  i  cos  6  +a,2  cos  20  +  03  cos  30  + .  .  . 

+  61  sin  0+&2  sin  20  +63  sin  30 +  .  .  .  ;       .     . 


(2) 


(3) 


or,  combining  the  sine  and  cosine  functions  by  the  binomial 
(par.  73), 

H    a0  +0  cos  (0-00  +c2cos  (20-/?2)  +c3  cos(30-/?3)+. . .  1 
a0  -  ci  sin  (0  -f  n)  +c2sin  (20  +  r2)  +c3  sin  (30  +  rs)  +. . .  J ' 

when1 

cn  =  Van2+bn2; 

bn 


tan  /?„ 


#   — . 


a». 


or 


On 

tan  rn=j—. 


.-,» 


The  proof  hereof  is  given  by  showing  that  the  coefficients 
an  and  h  n  of  the  series  (3)  can  be  determined  from  the  numerical 
values  of  the  periodic  function  (1),  thus, 

y=f(x)=Md) (6) 

Since,  however,  the  trigonometric  function,  and  therefore 
also  the  series  of  trigonometric  functions  (3)  is  univalent,  it 
follows  that  the  periodic  function  (6),  y=fo(0),  must  be  uni- 
valent, to  be  represented  by  a  trigonometric  series. 

77.  The  most  important  periodic  functions  in  electrical 
engineering  are  the  alternating  currents  and  e.m.fs.  Usually 
they  are,  in  first  approximation,  represented  by  a  single  trigo- 
nometric function,  as: 


i=  io  cos  (0-  to) 


or, 


e  =  en  sin  1  0—  o)- 
that  is.  they  arc  assumed  as  sine  waves. 


108  ENGINEERING  MATHEMATK  'S. 

Theoretically,  obviously  this  condition  can  never  be  perfectly 
attained,  and  frequently  the  deviation  from  sine  shape  is  suffi- 
cient to  require  practical  consideration,  especially  in  those  cases, 
where  the  electric  circuit  contains  electrostatic  capacity,  as  is 
for  instance,  the  case  with  long-distance  transmission  lines, 
underground  cable  systems,  high  potential  transformers,  etc. 

However,  no  matter  how  much  the  alternating  or  other 
periodic  wave  differs  from  simple  sine  shape — that  is,  however 
much  the  wave  is  "  distorted,"  it  can  always  be  represented 
by  the  trigonometric  series  (3). 

As  illustration  the  following  applications  of  the  trigo- 
nometric series  to  engineering  problems  may  be  considered : 

(A)  The  determination  of  the  equation  of  the  periodic 
function;  that  is,  the  evolution  of  the  constants  an  and  bn  of 
the  trigonometric  series,  if  the  numerical  values  of  the  periodic 
function  are  given.  Thus,  for  instance,  the  wave  of  an 
alternator  may  be  taken  by  oscillograph  or  wave-meter,  and 
by  measuring  from  the  oscillograph,  the  numerical  values  of 
the  periodic  function  are  derived  for  every  10  degrees,  or  every 
5  degrees,  or  every  degree,  depending  on  the  accuracy  required. 
The  problem  then  is,  from  the  numerical  values  of  the  wave, 
to  determine  its  equation.  While  the  oscillograph  shows  the 
shape  of  the  wave,  it  obviously  is  not  possible  therefrom  to 
calculate  other  quantities,  as  from  the  voltage  the  current 
under  given  circuit  conditions,  if  the  wave  shape  is  not  first 
represented  by  a  mathematical  expression.  It  therefore  is  of 
importance  in  engineering  to  translate  the  picture  or  the  table 
of  numerical  values  of  a  periodic  function  into  a  mathematical 
expression  thereof. 

(B)  If  one  of  the  engineering  quantities,  as  the  e.m.f.  of 
an  alternator  or  the  magnetic  flux  in  the  air-gap  of  an  electric 
machine,  is  given  as  a  general  periodic  function  in  the  form 
of  a  trigonometric  series,  to  determine  therefrom  other  engineer- 
ing quantities,  as  the  current,  the  generated  e.m.f.,  etc. 

A.  Evaluation  of  the  Constants  of  the  Trigonometric  Series  from 
the  Instantaneous  Values  of  the  Periodic  Function. 

78.  Assuming  that  the  numerical  values  of  a  univalent 
periodic  function  y=fo(0)  are  given;  that  is,  for  every  value 
of  11,  the  corresponding  value  of  y  is  known,  either  by  graphical 
representation,   Fig.    11;    or,  in  tabulated  form,  Table   I,  but 


TRIGONOMETRIC  SERIES. 


I<)«) 


the  equation  of  the  periodic  function  is  not  known.     It  can  be 
represented  in  the  form, 

y  =  uo  +  ax  cos  0+a2cos20H  a3cos30  +  .  .  .+aTOcosn0  +  .  .  . 

+  61  'sin  0  +  b2  sin  20+65  sin  30  +  ...+&„  sin  nO  +  .  .  .  ,    (7) 

and  the  problem  now  is,  to  determine   the  coefficients  a0,  ai, 

a-2 .  .  .  61,  62  •  •  •  • 


Fig.  41.     Periodifc  Functions. 
TABLE    I. 


fl 

V 

a 

V 

0 

y 

e 

V 

0 

-60 

90 

+  50 

ISO 

+  122 

270 

+  85 

10 

-49 

100 

+  61 

190 

+  124 

280 

+  65 

20 

-  38 

110 

+  71 

200 

+  126 

290 

+  35 

30 

-26 

120 

+  81 

210 

+  125 

300 

+  17 

40 

-12 

130 

+  90 

220 

+  123 

310 

0 

50 

0 

140 

+  99 

230 

+  120 

320 

-13 

60 

+  11 

150 

+  107 

240 

+  116 

330 

-26 

70 

+  27 

160 

+  114 

250 

+  110 

340 

-38 

80 

+  39 

170 

+  119 

260 

+  100' 

350 

-49 

90 

+  50 

180 

+  122 

270 

+  85 

360 

-60 

Integrate  the  equation  (7)  between  the  limits  0  and  2r.\ 
dO  +  ax  I  cos  OdO  +  a2  I      cos  20d0  + . 


JC'2r.  r2z  r2n  S*2Tt 

ydO  =  a0  I     dO  +  ca  \  cos  OdO  +  a2  I 
0  Jo  Jo  Jo 

+an  J      cos  nOdO  +  .  .  .+61  |     1 
sin  20d0  +  ...+/,„  J      s 


iOdO  +  .  .  .+&i  I     sin  0d0  + 


sin  20d0  + . .  .  +6n  I      sin  nOdO  + . 
„  /2"  ,        /sin  20  Z2" 

+0„/^ /*+...-&, /cos  s/2" 


n 


-6S 


/cos  20  /2*  .     /cos  w0  /2* 

/      2    /0  /      n    A, 


I  ID  ENGINEERING   MATHEMATICS. 

All  the  integrals  containing  trigonometric  functions  vanish, 
as  the  trigonometric  function  has  the  same  value  at  the  upper 
limit  2-  as  at  the  lower  limit  0,  that  is, 

/cos  nd  /2*     1  .        _  m     .. 

/ —  /     =-(cos  2mz— cos0)  =  0 

/      n    /o       wv 


/sin  nO  /2 
and  the  result  is 


-/     =-(sin  2?/--sin  ())=(), 
n     i  n        n 


"ydd  =  ao/o/    =2na0: 

,  (i 

hence 


f 


ao^C'ydO (8) 

-V" 
?/r/tf  is  an  element  of  the  area  of  the  curve  y,    Fig.  41,  and 

ydd  thus    is  the  area  of   the  periodic    function  y,  for   one 

period;  that  is, 

a0  =  ^zA, (9) 

where  A=  area  of  the  periodic  function  y=fo(6),  for  one  period; 
that  is,  from  0  =  0  to  0  =  2~. 

A 

2-  is  the  horizontal  width  of  this  area  A,  and  —  thus  is 

In 

the  area  divided  by  the  width  of  it;  that  is,  it  is  the  average 
height  of  the  area.  A  of  the  periodic  function  //;  or,  in  other 
words,  it  is  the  average  value  of  //.     Therefore, 

o0  =  avg.  (y)0 (10) 

The  first  coefficient,  a0,  thus,  is  the  average  value  of  the 
instantaneous  values  of  the  periodic  function  //,  between  0=0 
and  0  =  2-. 

Therefore,  averaging  the  values  of  y  in  Table  1.  gives  the 
first  constant  r;,,. 

79.  To  determine  the  coefficient  an,  multiply  equation  (7) 
by  COS  nd,  and  then  integrate  from  0  to  '_'-,  for  the  purpose  of 
making  the  trigonometric  functions  vanish.     This  gives 


TRIGONOMETRIC  SERIES.  Ill 

j      ycosnOdd     </o  j      cos  nihil)  \  a\  j      cos  ft0  cos  0d0 + 

Jo  Jil  ^t) 

+02  I      cos  ra0  cos  '20d0  +  .  .  . +a»  I     cos2  nOdO  +  .  .  . 

Vo  t/0 

+  hi  |     cosn0sin0d0+&2  I     cosn0sin20d0  +  . . . 

+&„  j      cos  ?i0  sin  nOdO  +  .  .  . 
Jo 

Hence,   by  the   trigonometric   equations  of  the   preceding 

section: 

(  ~  ycosnddd=a0  (    *cosn0d0+ai  |    |[cos(n+l)0+cos(n-l)0]d0 

Jo  Jo  Jo 

+  a,  I   "^[cos  (n+2)0+cos  (n-2)0]d0+. . . 
Jo 

+an(   *$(l+caa2n6)dd+..  . 

+5i  J  "'itsin  (n+ 1)0 -sin  (n-l)0]d0 

+62  I   ^[sin  (»+ 2)0- sin  (n-2)0]d0+. . . 
Jo 

+M     |sin2n0d0+.  .  . 

Jo 

All  those  integrals  of  trigonometric  functions  give  trigo- 
nometric functions,  and  therefore  vanish  between  the  limits  0 
and  2k,  and  there  only  remains  the  first  term  of  the  integral 
multiplied  with  an,  which  does  not  contain  a  trigonometric 
function,  and  thus  remains  finite: 

and  therefore, 

ri, 

I     y  cos  nddd=an7i:'} 

Jo 

hence 

aw=      (     //cos  n0d0 (U) 

71  Jo 


112 


ENGINEERING   MA  TUEMATK  'S. 


If  i he  instantaneous  values  of  y  are  multiplied  with  cos  nd, 
and  the  product  yn=ycos  nO  plotted  as  a  curve,  ycosnddd  is 
an  element  of  the  area  of  this  curve,  shown  for  ra  =  3  in  Fig.  42, 

f2* 

uml  thus    I     y  cos  nOdd  is  the  area  of  this  curve;  that  is, 


an  =  -A 


- 


nt 


(12) 


Fig.  42.     Curve  of  y  cos  30. 

where  An  is  the  area  of  the  curve  ?/cos  ft0,  between  0  =  0  ami 
0  =  2- 

As  2r  is  the  width  of  this  area  A„,  -^£  is  the  average  height 

of  this  area;   that  is.  is  the  average  value  of  y  cos  nO,  and  —  An 
thus  is  twice  the  average  value  of  y  cos  nd;  that  is, 

an  =  2  &vg.  (y  cos  n6)o2z (13) 


FlG.   !•!.     Curve  of  y  sill  30. 

The  coefficient  <i„  of  cos  n0  is  derived  by  multiplying  all 
the  instantaneous  values  of  y  by  cos  n0,  and  taking  twice  the 
average  of  the  instantaneous  values  of  this  product  //cos  nO. 


TRIGONOMETRIC  SERIES.  113 

80.  I>„  is  determined  in  the  analogous  manner  by  multiply- 
ing y  by  sin  nd  and  integrating  from  0  to  2;r;  by  the  area  of  (lie 
curve  //sin  nd}  shown  in  Fig.  43,  for  n  =  3, 

J// sin  nddd  =a,o  (     sin  nddd+ai  (      sin  nd  cos  0d0 
0  J)  Jo 

+  a>  (     sin  ra0  cos  26d0  +  .  .  .  H  a„  f     sin  n0 cos  nddd  +  .  .  . 
"Jo  Jo 

+  M     sin  n0  sin  0d0  +  62  |     sin  nft  sin  20d0  + .  . . 
Jo  Jo 

+bnl     sin2  nddd  +  ... 

ao|     sinn0d0  +  ai  )    i[sin  (n-!-l)0+sin  (n-l)0]d0 
+a2  J     i[sin  (n+2)0+sin  (n-2)0]d0  +  . . . 

+anl     \m\  2n0  10  + .  .  . 

Jo 

+  b  1  (      i[c<  »s  (to- 1)  0  -  cos  (to  + 1)  0]dd 

+&2  |J[cos  (to-2)0-cos  (to+2)0]J0  +  . . . 

+&„|     £[l-cos2n0]d0  +  ... 

-&«(     hd0  =  bn7t; 
Jo 

hence, 

&n=i  |    *ysin  nOdO (14) 

nJo 

=-An', (15) 

where  An'  is  the  area  of  the  curve  //,/=--// sin  nd,     Hence, 

bn  =  2  avg.  (//sin  v/^),,'",       (16) 


Ill 


ENGINEERING   MA  THEMATIC 'S. 


and  the  coefficienl  of  sin  nO  thus  is  derived  by  multiplying  the 
instantaneous  values  of  y  with  sin  nd,  and  then  averaging,  as 
twice  the  average  of  y  sin  nO. 

81.  Any  univalent  periodic  function,  of  which  the  numerical 
values  y  are  known,  can  thus  be  expressed  numerically  by  the 
eciuation, 
y=a0+ai  cos  0  +  a2  cos  2d  +  .  .  .  +an  cos  n0  +  .  .  . 


+  &i  sin  0  +  b2  sin  20  +  .  .  .  +  b„  sin  n0  +  . 


(17) 


where  the  coefficients  a0,  «i,  </L>,  .  .  .  &i,  &2  •  •  •  ,  are  calculated 
as  the  averages: 


2* 


a0=avg.  (y)0  ; 
ai=2avg.  {y  cos  fl^2*; 
a2  =  2  avg.  (?/  cos  20)o~' 
an  =  2  avg.  (?/  cos  nO)Q" 


&i=2  avg.  (//sin  0)o  *; 
62  =  2  avg.  (?/  sin  2#)0~  ; 
fr„  =  2  avg.  (y  sin  n0)o~n; 


•  (18) 


Hereby  any  individual  harmonic  can  be  calculated,  without 
calculating  the  preceding  harmonics. 

For  instance,  let  the  generator  c.m.f.  wave,  Fig.  44,  Table 
II,  column  23  be  impressed  upon  an  underground  cable  system 


Fig.  11.     Generator  e.m.f.  wave 


of  such  constants  (capacity  and  inductance),  that  the  natural 
frequency  of  the  system  is  670  cycles  per  second,  while  the 
generator  frequency  is  <><>  cycles.     The  natural  frequency  of  the 


TRIGONOMETRIC  SERIES. 


115 


cir<  nit  is  then  close  to  that  of  the  1 1th  harmonic  of  the  generator 
wave,  660  cycles,  and  if  the  generator  voltage  contains  an 
appreciable  11th  harmonic,  trouble  may  result  from  a  resonance 
rise  of  voltage  of  this  frequency;  therefore,  the  11th  harmonic 
of  the  generator  wave  is  to  he  determined,  that  is,  a\\  and  bu 
calculated,  but  the  other  harmonics  are  of  less  importance. 

Table  II 


0 

y 

cos  110 

sin  110 

;/  cos  11 II 

y  sin   1 1 9 

0 
10 
20 

5 

4' 

20 

+  1.000 
-0.342 

-0.700 

0 

+  0.940 

-  0 . 043 

+  5.0 

-1.4 

-15.3 

0 

+   3.8 
-12.9 

30 
40 
50 

22 

19 
25 

+  0.866 

+  0.174 
-0.985 

-0.500 
+  0.985 

-0.174 

+  19.1 

+  3.3 

-24.6 

-11.0 

+  1S.7 
-   4.3 

til) 
71) 

Ml 

29 
29 
30 

+  0.500 
+.0.643 

-0.940 

-  0 . 866 
+  0.700 
+  0 . 342 

+  14.5 

+  1S.0 
-2S.2 

-25.1 

+  22 . 2 
+ 10 . 3 

90 

]()() 
Hi) 

38 
46 
38 

0 
+  0.940 

-0.043 

-  1 . 000 
+  0.342 

+  0 . 766 

0 
+  43.3 

-24.4 

-3S.0 
+  15.7 

+  29  2 

120 
130 

140 

41 
50 
32 

-0.500 
+  0.985 

-0.174 

-0.866 
-0.174 

+  ().'.  is;, 

-  20 . 5 

+  49.2 
-5.0 

-35.5 

-8.7 
f31.5 

150 
160 

170 

30 
33 

7 

-  0 . 866 
+  0.700 
+  0.342 

0 .  500 
-0.643 

+  0.940 

-26.0 

+  25 . 3 

+  2.2 

-15.0 
-21.3 

ISO 

—  5 

Divided 

Total 

by  9 

+  34   5 
+  3.83     av 

-29.8 
-3.31=bn 

In  the  third  column  of  Table  II  thus  are  given  the  values 
of  cos  110,  in  the  fourth  column  sin  11 0,  in  the  fifth  column 
y  cos  110,  and  in  the  sixtli  column  y  sin  11^.  The  former  gives 
as  average  +1.915,  hence  a\\--=  +3. S3,  and  the  latter  gives  as 
average  -  1.655,  hence  &n  =  —3.31,  and  the  11th  harmonic  of 
the  generator  wave  is 

an  cos  110  4-6n  sin  110=3.83  cos  110-3.31  sin  110 

=  5.07  cos  (110 +41°) , 


HO  ENGINEERING  MATHEMATM  'S. 

hence.,  its  effective  value  is 

5.07 
-=-  =  3.58, 

V2 

while  the  effective  value  of  the  total  generator  wave,  that 
is.  the  square  root  of  the  mean  squares  of  the  instanta- 
neous values  y,  is 

e  =  30.5, 

thus  the  11th  harmonic  is  11.8  per  cent  of  the  total  voltage, 
and  whether  such  a  harmonic  is  safe  or  not,  can  now  be  deter- 
mined from  the  circuit  constants,  more  particularly  its  resist- 
ance. 

82.  In  general,  the  successive  harmonics  decrease;  that  is, 
with  increasing  11,  the  values  of  a„  and  bn  become  smaller,  and 
when  calculating  an  and  bn  by  equation  (18),  for  higher  values 
of  n  they  arc  derived  as  the  small  averages  of  a  number  of 
large  quantities,  and  the  calculation  then  becomes  incon- 
venient and  less  correct. 

Where  the  entire  series  of  coefficients  an  and  b„  is  to  be 
calculated,  it  thus  is  preferable  not  to  use  the  complete  periodic 
function  y,  but  only  the  residual  left  after  subtracting  the 
harmonics  which  have  already  been  calculated;  that  is,  after 
a0  has  been  calculated,  it  is  subtracted  from  y,  and  the  differ- 
ence, ?/j  =y  —  (i(),  is  used  for  the  calculation  of  (i\  and  &i. 

Then  a,\  cos  0+&i  sin  0  is  subtracted  from  yl}  and  the 
difference, 

ll'z  =  y\—(a\  cos  0+h\  sin  0) 
=  y—(a0+aicox  0+b-i  sin  0), 

is  used  for  the  calculation  of  a2  and  b2. 

Then  <i2  cos  20+&2sin  28  is  subtracted  from  y2)  and  the  rest, 
//:i,  used  for  the  calculation  of  a:i  and  &3,  etc. 

In  this  manner  a  higher  accuracy  is  derived,  and  the  calcu- 
lation simplified  by  having  the  instantaneous  values  of  the 
function  of  the  same  magnitude  as  the  coefficients  a„  and  l>„. 

As  illustration,  is  given  in  Table  III  the  calculation  of  the 
first  three  harmonics  of  the  pulsating  current,  Fig.  41,  Table  I: 


TRIGONOMETRIC  SERIES.  117 

83.  In  electrical  engineering,  the  mosl  important  periodic 
('unctions  arc  the  alternating  currents  and  voltages.  Due  to 
the  constructive  features  of  alternating-current  generators, 
alternating  voltages  and  currents  are  almost  always  symmet- 
rical waves:  that  is,  the  periodic  function  consists  of  alternate 
half-waves,  which  are  the  same  in  shape,  but  opposite  in  direc- 
tion, or  in  other  words,  the  instantaneous  values  from  180  cleg. 
to  ."><>()  deg.  are  the  same  numerically,  but  opposite  in  sign, 
from  the  instantaneous  values  between  0  to  180  deg.,  and  each 
cycle  or  period  thus  consists  of  two  equal  but  opposite  half 
cycles,  as  shown  in  Fig.  44.  In  the  earlier  days  of  -  electrical 
engineering,  the  frequency  has  for  this  reason  frequently  been 
expressed  by  the  number  of  half-waves  or  alternations. 

In  a  symmetrical  wave,  those  harmonics  which  produce  a 
difference  in  the  shape  of  the  positive  and  the  negative  half- 
wave,  cannot  exist;  that  is,  their  coefficients  a  and  b  must  be 
zero.  Only  those  harmonics  can  exist  in  which  an  increase  of 
the  angle  6  by  180  (leg.,  or  -,  reverses  the  sign  of  the  function. 
This  is  the  case  with  cos  nO  and  sin  nd,  if  n  is  an  odd  number. 
If,  however,  n  is  an  even  number,  an  increase  of  0  by  ~  increases 
the  angle  nO  by  2-  or  a  multiple  thereof,  thus  leaves  cos  nd 
and  sin  nd  with  the  same  sign.  The  same  applies  to  a0.  There- 
fore, symmetrical  alternating  waves  comprise  only  the  odd 
harmonics,  but  do  not  contain  even  harmonics  or  a  constant 
term,  and  thus  are  represented  by 

y  =  a\  cos  0  +  a3  c< >s  30  +  a5  cos  50  +  .  .  . 
+  6,  sin  d+b3  sin  30 +b5  sin  50  + (19) 

When  calculating  the  coefficients  a„  and  h„  of  a  symmetrical 
wave  by  the  expression  (18),  it  is  sufficient  to  average  from  0 
to  -;  that  is,  over  one  half-wave  only.  In  the  second  half-wave, 
cos  nd  and  sin  nd  have  the  opposite  sign  as  in  the  first  half-wave, 
if  n  is  an  odd  number,  and  since  y  also  has  the  opposite  sign 
in  the  second  half-wave,  y  cos  nO  and  //sin  nd  in  the  second 
half-wave  traverses  again  the  same  values,  with  the  same  sign, 
as  in  the  first  half-wave,  and  their  average  thus  is  given  by 
averaging  over  one  half-wave  only. 

Therefore,  a  symmetrical  univalent   periodic  function,  as  an 


lis 


ESCIXEERIXC  MATHEMATIt  'S. 


Table 


e 

y 

yi=y-oa 

j/,  cos  0 

yx  sin  0 

Cl  =  «l  cosfl 
+  6i  sin  0 

U2=yn—c. 
I 

0 
10 

20 

-60 
-49 
-38 

-111 

-100 

-89 

-111 

-98 

-84 

0 
-17 
-30 

-84 
-85 
-83 

-27 
-15 

-6 

30 
40 
50 

-26 
-12 

0 

—  77 
-63 
-51 

-67 
-48 
-33 

-38 
-40 
-39 

-79 
-72 
-63 

+  2 

9 

12 

60 

70 
SO 

+  11 
27 
39 

-40 
-24 
-12 

-20 
-8 

—  2 

-35 
-23 
-12 

-52 
-40 
-26 

12 

16 

14 

90 
100 

110 

61 

71 

-1 

+  10 

20 

0 
_2 

-7 

-1 
+  10 
+  19 

-11 

+  4 

IS 

10 
6 

+  2 

120 
130 
140 

81 
!IO 
99 

30 
39 

48 

-15 

-37 

+  26 

+  30 
+  31 

32 
45 

58 

_2 

-6 

-10 

150 
160 

170 

107 
114 
119 

56 
63 

6S 

-49 
-59 
-67 

+  28 
+  22 
+  12 

67 
75 
81 

-11 
-12 
-13 

L80 
190 

200 

122 

124 
126 

71 
73 
75 

-71 
-72 
-71 

0 
-13 
-26 

84 
85 
83 

-13 
-12 

-8 

210 
220 
230 

125 
123 

120 

71 
72 
69 

-64 
—  55 

-44 

-37 
-47 
-53 

79 
72 
63 

—  5 
0 

+  6 

240 
250 

260 

116 

110 
100 

65 
59 
49 

-32 

_90 

-9 

-28 
-56 

-48 

52 
40 
26 

13 

19 
23 

270 
280 
290 

85 
65 
35 

34 
+  14 
-16 

0 
+  2 
-5 

-34 
-14 
+  15 

11 

-4 

-18 

23 

18 

.   +2 

300 
310 
320 

+  17 

0 

-13 

-34 
-51 
-64 

-17 
-33 
T49 

+  30 
+39 

+  41 

-32 

-  15 

-  58 

_2 
-6 

-6 

330 
340 
350 

-26 
-38 
-49 

—  75 

-89 

-100 

-65 

-St 

-99 

+  37 
+  30 
+  17 

-67 
—  75 
-81 

-8 
-14 
-19 

Total  •  • 
1  divided 

by  36  ... 

+  1826 
+  50.7=o„ 

Total 
Divided  by 

is 

.-1520 

-S4.4  =  a, 

-204 
-11.3  =  6, 

Total 
Divided 

by  18... 

TRIGi  )NOM  ETRK '  SERIES. 


\\) 


III. 


C,=(11  COS  2( 

Vi  C03  20 

-27 

!/..  sin  29 

'+62sin2tf 

V3  =  V2—Ci 

l/3  cos  30 

}/.,  sin  3D 

0 

0 

-15 

-12 

-12 

0 

0 

-14 

—  5 

-12 

-3 

-3 

-1 

10 

—  5 

-4 

-7 

+  1 

0 

+  1 

20 

+  1 

+  2 

-1 

+  3 

0 

+  3 

30 

+  2 

+  9 

+  4 

+  5 

2 

+  4 

40 

_2 

+  12 

11 

+  1 

-1 

0 

50 

6 

+  10 

13 

-1 

+  1 

0 

60 

12 

+  10 

15 

+  1 

-1 

0 

70 

13 

+  5 

10 

_2 

+  1 

+  2 

SO 

-10 

0 

15 

-5 

0 

+  5 

00 

-6 

—  2 

12 

-0 

-3 

+  5 

100 

_2 

-1 

7 

—  5 

-4 

+  2 

110 

+  1 

+  2 

+  1 

-3 

-3 

0 

120 

+  1 

+  6 

-4 

_2 

-1 

130 

_2 

+  10 

-11 

+  1 

0 

+  1 

110 

—  5 

+  10 

-13 

+  2 

0 

+  2 

150 

-9 

+  8 

-15 

+  3 

-1 

+  3 

160 

-12 

-4 

-10 

+  3 

-3 

+  1 

170 

-13 

0 

-15 

+  2 

_2 

0 

ISO 

-11 

-4 

-12 

0 

0 

0 

100 

-6 

-0 

-7 

-1 

0 

-1 

200 

_2 

-4 

-1 

-4 

0 

-4 

210 

0 

0 

+  4 

-4 

—  2 

-4 

220 

-1 

+  6 

11 

—  5 

-4 

_2 

230 

-6 

+  11 

13 

0 

0 

0 

240 

-15 

+  12  * 

15 

+  4 

+  4 

+  2 

250 

—  22 

+  8 

16 

+  7 

+  3 

+  6 

260 

-  23 

0 

15 

+  8 

0 

+  8 

270 

-17 

-6 

12 

+  0 

-3 

+  5 

280 

—  2 

-1 

7 

-5 

+  4 

_2 

290 

+  1 

+  2 

+  1 

-3 

+  3 

0 

300 

+  1 

+  6 

-4 

_2 

+  2 

+  1 

310 

-1 

+  6 

-11 

+  5 

_2 

-4 

320 

-4 

+  7 

-13 

+  5 

0 

—  5 

330 

-11 

+  9 

-15 

+  1 

0 

-1 

340 

-IS 

+  6 

-10 

-3 

-3 

+  1 

350 

-270 

+  120 

Total 

-33 

+  27 

-15.0  =  a. 

+  0 . 7  =  b.. 

Divided  by  IS 

-1.8  =  a3 

+  l.o  =  b. 

[20 


ENGINEERING  MATHEMATR  'S. 


alternating  voltage  and  current  usually  is,  can  be  represented 
by  the  expression, 

y  =  al  cos  6+0,3  cos  3  d+a5  cos  5  0+a7  cos  70+.  .  . 

bj  sin  0  ■  //.;  sin  3  0  +  b5  sin  5  0  +  b7  sin  7  0  +.  .  .:       (20; 


where, 

e,     2  avg.  (y  cos  0)0"; 
a3    2  avg.  i.y  cos  30)o*; 
a.-,    2  avg.  i.y  cos  5^)0"; 
u7  =  2  avg.  (//  cos  70)0'; 


bi  =  2  avg.  i//  sin  0)o*; 
63=2  avg.  (v  sin  36)*; 
&5  =  2avg.  (ysin50)o*;    j 

67  =  2  avg.  (//  sin  70)J\    j 


(21) 


84.  From  180  cleg,  to  300  deg.,  the  even  harmonics  have 
the  same,  but  the  odd  harmonics  the  opposite  sign  as  from  0 
to  ISO  deg.  Therefore  adding  the  numerical  values  in  the 
range  from  ISO  deg.  to  360  deg.  to  those  in  the  range  from  0 
to  ISO  deg.,  the  odd  harmonics  cancel,  and  only  the  even  har- 
monics remain.  Inversely,  by  subtracting,  the  even  harmonics 
cancel,  and  the  odd  ones  remain. 

Hereby  the  odd  and  the  even  harmonics  can  be  separated. 
If  y  =  y{6)  arc  the  numerical  values  of  a  periodic  function 
from  0  to  ISO  deg.,  and  y'  =  y(6+Tz)  the  numerical  values  of 
the  same  function  from  180  deg.  to  300  deg., 

y2{d)  =  h{y{6)+y(6+n)),  ....  (22) 
is  a  periodic  function  containing  only  the  even  harmonics,  and 

yi(3)  =  h\y(0)-y(d+7z)} (23) 

is  a  periodic  function  containing  only  the  odd  harmonics;  that  is: 

yi{0)     <l\  COS  0+(l;\  cos  30+  as  cos  50  I  .  .  . 

+bi  sin  0  +  b3  sin  3  Q+b5  sin  50  +  . . .;      .     .     (24) 

y2i  0 )  =  ao+a2  cos  20  +u4  cos  46  +  . .  . 

/»,  sin  26  l-54sin  40  + (25) 

and  the  complete  function  is 

2/(0)     yi{6)  Vy2{6) (26) 


TRIGONOMETRIC  SERIES.  121 

By  this  method  it  is  convenient  to  determine  whether  even 
harmonics  are  present,  and  if  they  are  present,  to  separate 
them  from  the  odd  harmonics. 

Before  separating  the  even  harmonics  and  the  odd  har- 
monics, it  is  usually  convenient  to  separate  the  constant  term 
</o  from  the  periodic  function  y,  by  averaging  the  instantaneous 
values  of  ll  from  0  to  360  deg.  The  average  then  gives  a0, 
and  subtracted  from  the  instantaneous  values  of  y,  gives 

yo(0)=y(0)~ao (27) 

as  the  instantaneous  values  of  the  alternating  component  of  the 
periodic  function;  that  is,  the  component  y0  contains  only  the 
trigonometric  functions,  but  not  the  constant  term.  y0  is 
then  resolved  into  the  odd  series  //i,  and  the  even  series  y2. 
85.  The  alternating  wave  y0  consists  of  the  cosine  components : 

u(d)  =  a\  cos  0  +  a2  cos  20+a3  cos  30  +  a4  cos  40  +  .  . .,    (28) 

and  the  sine  components: 

v(0)=bi  sin  0  +  b2  sin  20 +63  sin  30  +  h4  sin  40'+. . .;    (29) 

that  is, 

yo{6)=u(6)+v(d) (30) 

The  cosine  functions  retain  the  same  sign  for  negative 
angles  (—0).  as  for  positive  angles(  +0),  while  the  sine  functions 
reverse  their  sign;  that  is, 

w(-0)=+m(0)     and     i'(-0)  =    -v{0).       .     .     .     (31) 

Therefore,  if  the  values  of  y0  for  positive  and  for  negative 
angles  0  are  averaged,  the  sine  functions  cancel,  and  only  the 
cosine  functions  remain,  while  by  subtracting  the  values  of 
y0  for  positive  and  for  negative  angles,  only  the  sine  functions 
remain;   that  is, 

y0(d)+yQ(-d)=2u(d),' 

■ (32) 

>/o(0)-//o(-0)=2e(0);_ 

hence,  the  cosine  terms  and  the  sine  terms  can  be  separated 
from  each  other  by  combining  the  instantaneous  values  of  yQ 
for  positive  angle  II  and  for  negative  angle  (—0),  thus: 

u(6)  =  iiy0(d)+yo(-d)\, 


v(e)  =  i\y0(d)-y0(-d)}. 


(33) 


1 22  ENGINEERING   M .  1 77/  EM  A  Til 'S. 

Usually,  before  separating  the  cosine  and  the  sine  terms, 
u  and  v,  first  the  constant  term  a0  is  separated,  as  discussed 
above;  that  is.  the  alternating  function  y0  =  y— a0  used.  If 
the  general  periodic  function  y  is  used  in  equation  (33),  the 
constant  term  a{)  of  this  periodic  function  appears  in  the  cosine 
term  >/,  thus: 

u{0)  =  \ \y{6)  +  y (- 0)]  =  a0 +ai  cos  0  +  a2  cos  20  +  as  cos  30  + .  .  ., 

while  r(0)  remains  the  same  as  when  using  y0. 

86.  Before  separating  the  alternating  function  y0  into  the 
cosine  function  u  and  the  sine  function  r,  it  usually  is  more 
convenient  to  resolve  the  alternating  function  y0  into  the  odd 
series  //i,  and  the  even  series  y2)  as  discussed  in  the  preceding 
paragraph,  and  then  to  separate  y\  and  //_>  each  into  the  cosine 
and  the  sine  terms: 

ui(0)  =  }i\y1(0)  +  y1(-0)\=aicos0+a:ivos:]0  |  -a5cos50d  .  .  .;  1 

\  (34) 
vi(8)=  \{yx{d)    f/i(-0)}=&isin0+&3sin  30+&5sin  50+. .  . 


?/_,.  0)      \)  //,i  0)  +y2{-6)  \  =  a2  cos  20  \  a ,  cos  10  +  .  . .;  ] 

v-AO)     l;//,i//i   ->j2(-0)\=b2mi  20+b4  sin  40  +  .  .  . 


(35) 


In  the  odd  functions  u\  and  Vi,  a  change  from  the  negative 
angle  (—  6)  to  the  supplementary  angle  {n—d)  changes  the  angle 
of  the  trigonometric  function  by  an  odd  multiple  of  x  or  ISO 
deg.,  that  is,  by  a  multiple  of  2-  or  360  deg.,  plus  ISO  deg., 

which  signifies  a  reversal  of  the  function,  thus: 

um=h\!jl(0)-!/l^-0)\,  i 

.      .     .     .     (30) 
vi(0)  =  i\yi(d)+yi(7t-0)}.  \ 

However,  in  the  even  functions  n2  and  v2  a  change  from  the 
negative  angle  (  — 0)  to  the  supplementary  angle  (n     0),  changes 

the  angles  of  the  trigonometric  function  by  an  even  multiple 
of  ~:  that  is.  by  a  multiple  of  2n  or  360  d^j:.:  hence  leaves 
the  sign  of  the  trigonometric  function  unchanged,  thus: 

0)      \{y2{6)  Yy2{n-  0)},  1 

.      .     .     .     (37) 
v2{6)  =  \\  y2(0)  -y2(n     0)\.\ 


TRIGONOMETRIC  SERIES. 


123 


To  avoid  the  possibility  of  a  mistake,  it  is  preferable  to  use 

the  relations  (.'I  I )  and  (35),  which  are  the  same  for  the  odd  and 
for  the  even  series. 

87.  Obviously,  in  the  calculation  of  the  constants  an  and 
b„,  instead  of  averaging  from  0  to  180  deg.,  the  average  can 
be  made  from  -90  deg.  to  +90  deg.  In  the  cosine  function 
u(0),  however,  the  same  numerical  values  repeated  with  the 
same  signs,  from  0  to  —90  deg.,  as  from  0  to  +90  deg.,  and 
the  multipliers  cos  nO  also  have  the  same  signs  and  the  same 
numerical  values  from  0  to  —90  deg.,  as  from  0  to  +90  deg. 
In  the  sine  function,  the  same  numerical  values  repeat  from  0 
to  —90  deg.,  as  from  0  to  +90  deg.,  but  with  reversed  signs, 
and  the  multipliers  sin  nO  also  have  the  same  numerical  values, 
but  with,  reversed  sign,  from  0  to  —90  deg.,  as  from  0  to  +90 
deg.  The  products  u  cos  nO  and  v  sin  nO  thus  traverse  the 
same  numerical  values  with  the  same  signs,  between  0  and 
-90  deg.,  as  between  0  and    +90  deg.,  and  for  deriving  the 


TZ 


averages,  it  thus  is  sufficient  to  average  only  from  0  to  — ,  or 

— ' 

90  deg.:   that  is,  over  one  quandrant. 

Therefore,  by  resolving  the  periodic  function  y  into  the 
cosine  components  u  and  the  sine  components  v,  the  calculation 
of  the  constants  o„  and  bn  is  greatly  simplified;  that  is,  instead 
of  averaging  over  one  entire  period,  or  360  deg.,  it  is  necessary 
to  average  over  only  90  deg.,  thus: 


ai=2  avg.  (u\  cos  0)o2  ;  &i=2  avg.  {vj  sin  6)Q2  ; 

■k  r. 

a2  =  2  avg.  (  U2  cos  20)o 2  ;  h2  =  2  avg.  (v2  sin  20)o- : 

a.3  =  2  avg.  (u-3  cos  30)o2  '.  &3=2  avg.  (r:>,  sin  3#)o2  ; 

-  r 

fl4  =  2  avg.  ( // 1  cos  40)o2  ;  64  =  2  avg.  (r4  sin  40)02  ; 


"5=2 


avg.  (  //.-,  cos  .V/ In- 
ch'. 


?>5  =  2  avg.  (  V5  sin  50)o2 
etc. 


(38) 


where  u\  is  the  cosine  term  of  the  odd  function  2/1;  u2  the 
cosine  term  of  the  even  function  //2;  U3  is  the  cosine  term  of 
the  odd  function,  after  subtracting  the  term  with  cos  //;  that  is, 

U3=ui— ai  cos  0, 


L2  1  ENGINEERING   MATHEMATICS. 

analogously,  "i  is  the  cosine  term  of  the  even  function,  after 
subtracting  the  term  cos  20; 

U4  =  U2  —  Cl2  COS  20, 

and  in  the  same  manner, 

U5=V,3  —  ds  COP  30, 

u6  =  uA  —  <u  cos  40, 

and  so  forth;    V\,  r2,   r.-,,   i>4,  etc.,  arc  the  corresponding  sine 
terms. 

When  calculating  the  coefficients  a„  and  bn  by  averaging  over 
90  deg.,  or  over  180  deg.  or  360  deg.,  it  must  be  kept  in  mind 
that  the  terminal  values  of  y  respectively  of  u  or  v,  that  is, 
the  values  for  0=0  and  0  =  90  deg.  (or  0  =  180  deg.  or  360 
deg.  respectively)  are  to  be  taken  as  one-half  only,  since  they 
are  the  ends  of  the  measured  area  of  the  curves  an  cos  nO  and 
b„  sin  nil,  which  area  gives  as  twice  its  average  height  the  values 
an  and  b„,  as  discussed  in  the  preceding. 

In  resolving  an  empirical  periodic  function  into  a  trigono- 
metric series,  just  as  in  most  engineering  calculations,  the 
most  important  pari  is  to  arrange  the  work  so  as  to  derive  the 
results  expeditiously  and  rapidly,  and  at  the  same  time 
accurately.  By  proceeding,  for  instance,  immediately  by  the 
general  method,  ('([nations  (17)  and  (18),  the  work  becomes  so 
extensive  as  to  be  a  serious  waste  of  time,  while  by  the  system- 
atic resolution  into  simpler  functions  the  work  can  be  greatly 
reduced. 

88.  In  resolving  a  general  periodic  function  y(d)  into  a 
trigonometric  -cries,  the  most  convenient  arrangement  is: 

1.  To  separate  the  constant  term  a0,  by  averaging  all  the 
instantaneous  values  of  y(6)  from  0  to  360  deg.  (counting  the 
end  values  at  0  =  0  and  at  0  =  360  deg.  one  half,  as  discussed 
above) : 

a0  =  avg.  [y(d)}02* (10) 

and  then  subtracting  (/,,  from   1/(11),  gives   the  alternating  func- 
tion, 

yo(0)=y(0)-ao. 


TRIGONOMETRIC  SERIES. 


125 


2.  To   resolve   the   general  alternating  function   yo(0)   into 
the  odd  function  y\{0),  and  the  even  function  1/2(0), 

//,(//)     J{yo(0)  -  //()(V/+^)j;      ....     (23) 
2/2(0)  -*{%(*)  +2/o(0+tt)1 (22) 

3.  To  resolve  yi(0)  gnd  //-(^n  into  the  cosine  terms  u  and 

the  sine  terms  v, 

ui(0)  =  ${yl(d)+y1(-d)\:  } 

......     (34) 

Vi(e)=i{yi(d)-yi(-6)}: 


(35) 


;  &i,  h-2,  b3. 


U2(0)  =  h\y2(0)+uA -")[: 
V2{0)  =  hiy2(0)-y2(-0)\.  j 

4.  To  calculate   the  constants  «i,   fl2,  «:j 

bv  the  averages, 

J£   1 
a  „  =  2  avg.  (w^cos  n0)o  2  ;  | 

-    I  ' 
6n  =  2  avg.(v„sm  n0)o2.  j 

If  the  periodic  function  is  known  to  contain  no  even  har- 
monics, that  is,  is  a  symmetrical  alternating  wave,  steps  1  and 
2  are  omitted. 


I'm;.  1.").     Mean  Daily  Temperature  at  Schenectady. 

89.  As  illustration  of  the  resolution  of  a  general  periodic 
wave  may  he  shown  the  resolution  of  the  observed  mean  daily 
temperatures  of  Schenectady  throughout  the  year,  as  shown 
in  Fig.  45,  up  to  the  7th  harmonic* 


*  The  numerical  values  of  temperature  cannot  claim  any  great  absolute 
accuracy,  as  they  are  averaged  over  a  relatively  small  number  of  years  only, 
and  observed  by  instruments  of  only  moderate  accuracy.  For  the  purpose 
of  illustrating  the  resolution  of  the  empirical  curve  into  a  trigonometric 
series,  this  is  not  essential,  however. 


126 


ENG1  VEERING   MA TI1KM.\TH 'S. 


Table  IV 


(i) 
e 

u 

(3) 
y  —  no  =  J/o 

(I) 

Z/l 

(5) 

Jan. 

0 

10 
2(1 

-   4.2 

-  4.7 

-  5.2 

-12.95 
-13.45 
-13.95 

-13.10 
-13.55 
-13.65 

+  0.15 

+  0.10 
-0.30 

Feb. 

30 
40 

50 

-    5.4 

3  .  S 
2.0 

-14.15 
- 12 . 55 
-11.35 

-13.55 

.  -12.35 

-11.20 

-0.60 
-0.20 
-0.15 

Mar. 

00 
70 

SO 

1.6 
+   0.2 

+    l.S 

-10.35 

-    S.55 

-   0.95 

-  9.75 
7.65 

-  6.05 

-0.60 
-0.90 
-0.90 

Apr. 

DO 
100 

111) 

+   5.1 
+    9.1 
+  11.5 

-    3.65 
+   0.35 
+   2.75 

-  3.35 

-  0.35 
+    1.75 

-0.30 
+  0.70 
+  1.00 

May 

120 

130 
110 

+  13.3 

+  15.2 

+  17.7 

+    4.55 
+   6.45 

+    S.95 

+   3.90 
+   5.85 
+   S.15 

+  0.65 
+  0.60 
+  0.80 

June 

150 
100 
170 

+  19.2 
+  19.5 
+  20.6 

+  10.45 
+  10.75 
+  11.85 

+  10.10 
+  10.80 
+  12.15 

+  0.35 
-0.05 
-0.30 

July 

ISO 

190 

200 

+  22.0 

+  22.4 
+  22.1 

+  13.25 
+  13.65 

+  13.35 

Aug. 

210 
220 
230 

+  21.7 
+  20.9 
+  19.S 

+  12.95 
+  12.15 
+  11.05 

Sept. 

240 

250 
200 

+  17.9 
+  15.5 
+  13.S 

+    9.15 
+    0.75 
+    5.15 

Oct. 

270 
2S0 
2!  10 

+  11:8 

+    9.S 
+   S.O 

+    3.05 
+    1.05 
-   0.75 

Nov. 

300 

310 
320 

+    5.5 
+   3.5 
+    1.4 

-  3.25 
■    5.25 

-  7.35 

Dec. 

330 

310 
350 

1.0 

-  2.1 

-  3.7 

-    9.75 
-10.85 

12.  15 

Total 
Dividei 

1  by  36. 

315.1 

S.75  =  r/„ 

TRIGONOMETRIC  SERIES. 


127 


Table  V. 


(1) 

0 

(2) 
Vl 

(3) 

HI 

i 

Vl 

(5) 

1/2 

(6) 
m 

(7) 

-  90 

SI) 

-70 

+    3.35 
+    0.35 
-    1.75 

-0.30 
+  0.70 
+  1.00 

+  0.(15 
+  0.(10 
+  0.80 

, 

lid 
-50 
-40 

-  3.90 
5  85 

-   8.15 

-3.0 
-20 

-10 

0 

+  10 

+  20 

-10.10 
-10.80 
-12.15 

-13.10 
-13.55 
-13.05 

+  0.35 
-0.05 
-0.30 

+  0.15 
+  0.10 
-0.30 

+  0.15 
-0.10 
-0.17 

0 
+  0.20 
-0.12 

-13.10 
-12.85 
-12.23 

0 
-0.70 
-1.42 

+  30 
+  40 
+  50 

-13.55 
-12.35 
-11.20 

-11.82 
-10.25 

-   8.53 

-1.73 

-2.10 

-2.67 

-0.60 
-0.20 
-0.15 

-0.12 
+  0.30 
+  0.22 

-0.47 
-0.50 
-0.37 

+  00 
+  70 

+  S0 

-  9.75 
7.(15 

-  G.05 

-  6.82 

-   4.70 

-  2.85 

-  2 .  93 
-2.95 
-3.20 

-0.60 
-0.90 
-0.90 

+  0.02 
+  0.05 
-0.10 

-0.62 
-0.95 
-0.80 

+  90 

-   3.35 

0 

-  3 .  35 

-0.30 

-0.30   • 

0 

L28 


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'0 

CO 

|  -a 

y; 

C"-     *     i- ' 

/- 

© 

© 

•-vj 

Q 

IN 

u 

= 

h- 

co 

>e       —      ,^ 

CI    C 

© 

1  - 

T 

^ 

01 

© 
01 

oi 

01     CO     CO 

^" 

-1 

1          1          1 

1 

1 

1 

1 

1 

1 

!      1      1 

—    6 

o    c^ 

^ 

~ 



^ 

, . 

. 

_ 

. .    

•  —  ^ 

t— «  ^ 

'•>*        r* 

i— l 

01 

C3 

-7- 

1 ~ 

to 

\~    s    ~ 

■~ 

•  ^  <        — ' 

■- 

G  S 

i:j() 


ENGINEERING   MA  THEMA  TI<  'S. 
Table  VIII. 

COSIXE   SERIES   w2. 


(1) 

6 

2 
Hi 

(3) 

hi  cos  20 

1 
as  ci  is  26 

(5) 
in 

(6) 
in  cos  40 

(7) 
n 4  cos  4  0 

(8) 
lit 

its  cos  60 

0 

111 

20 
30 

to 

50 

60 
70 
80 
90 

+  0.15 
-0.10 
-0.17 

-0.12 
|  0.30 

+  0 .  22 

+  0 .  02 
+  0.05 
-0.10 
-0.30 

K  +  0.15) 
-0.09 
-0.13 

-0.06 
+  0.05 

-0.04 

-0.01 
-0.04 
+  0.09 
K  +  0.30) 

0 

0 

+  0.15 
-0.10 

-0.17 

-0.12 
+  0.30 
+  0.22 

+  0 .  02 
+  0.05 
-0.10 
-0.30 

K+0.15) 

-o.os 

-0.03 

+  0.06 
-0.29 
-0.21 

-0.01 
+  0.01 
-0.08 
K  +  0.30) 

-0.16 
-0.12 

-0.03 

+  0.08 
+  0.15 
+  0.15 

+  0.08 
-  0 .  03 
-0.12 
-0.16 

+  0.31 
+  0.02 
-0.14 

-0.20 
+  0.15 
+  0.07 

-0.06 
+  0.08 
+  0.02 
-0.14 

K+0.31) 
+  0.01 
+  0 .  07 

+  0 .  20 
-0.07 
+  0.03 

-0.06 
+  0.04 
-0.01 

K  +  0.14) 

Total 

Divided  by 

(.i 

Multiplied 

by  2.... 

-0.01 
-0.001 

-0.002 

=  a2 

-0.71 

-0.079 

-0.15S 
=  a3 

+  0.44 
+  0.049 
+  0.09S 

Table  IX. 

STXE   SERIES   v2. 


(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

(«) 

6 

Vi 

Vi  ^in  20 

b-i  sin  20 

V4 

04  sin  40 

64  sin  40 

J-6 

re  sin  GO 

0 

0 

0 

10 

+  0.20 

+  0.07 

-0.20 

+  0.40 

+  0.26 

+  0.22 

+  0.1S 

+  0.16 

20 

-0.12 

-0.08 

-0.39 

+  0.27 

+  0.27 

+  0.34 

-0.07 

-0.07 

30 

-0.47 

-0.41 

-0.52 

+  0.05 

+  0.04 

+  0.30 

-0.25 

+  0 

10 

-0.50 

-0.49 

-0.59 

+  0.09 

+  0.03 

+  0.12 

-0.03 

+  0.0:', 

50 

-0.37 

-0.36 

-0.59 

+  0.22 

-O.os 

-0.12 

+  0.34 

-0.30 

60 

0.62 

-  0.51 

-0.52 

-0.10 

+  0.09 

-0.30 

+  0.20 

0 

70 

0.95 

-0.61 

-0.39 

-0.56 

+  0.55 

-0.34 

-0.22 

-0.19 

SO 

0.80 

■0.27 

0.20 

-0.60 

+  0.39 

+  0.22 

-0.38 

0.:;:; 

90 

0 

0 

Tota 

1 

2.69 

+ 1 .  :^ 

-0.70 

Divid 

ed  by  ' 

0.30 

+  0.172 

-0.07s 

Divid 

ed   by  2 

0.60 
b2 

;  ().'!!  1 
=  *>< 

-0.156 
=6. 

TRIGONOMETRK '  SERIES.  1 3 1 

Tabic  IV  gives  the  resolution  of  the  periodic  temperature 
function  into  the  constant  term  «„,  the  odd  series  Mi  and  the 
even  series  }i-i. 

Table  V  gives  the  resolution  of  the  series  ij\  and  y2  into 
the  cosine  and  sine  series  Mi,  Wi,  u-2,  v2. 

Tables  VI  to  IX  give  the  resolutions  of  the  series  Mi,  Vi,  u2, 
/-,  and  thereby  the  calculation  of  the  constants  an  and  b„. 

90.  The  resolution  of  the  temperature  wave,  up  to  the 
7th  harmonic,  thus  gives  the  coefficients: 

a0=  +8.75; 

01  = -13.28;  61  =  -3.33; 

a2= -0.001;  &2= -0.602; 

a3=-0.33;  63=  -0.14; 

a4= -0.154;  &4=  +0.386 

a5=  +0.014;  &5  =  -0.090 

a6= +0.100;  &6=  -0.154 

a7  =  -0.022;  67  =  -0.0S2 

or,    transforming    by    the    binomial,    ancosnO+bnsmnd  =  cncos 

(nO—j-n),  by  substituting  cn  =  Van2  +  bn2  andtan?-re=—  gives, 

a0=+8.75; 

ci=-13.69;     n=+14.150;  or  ri=+U.15°; 

c2= -0.602;     r2=+S9.9°;     or  ^=+44.95°+180n; 

c3=+0.359;     r3=-23.0°;     or  ^=-7.7+120n=+112.3+120m; 

c4=-0.416;     r4=-68.2°;    or  ^4=-17.05+90w=+72.95+90m; 

4 

c5=+0.091;     r5=-Sl-15°;  or  ^5=  - 16.23 +72n=  +55.77+ 72m; 
c6=+0.184;     r6=-57.0°;    or  j    -9.5+60n= +50.5+60m; 

c7=-0.085;     r7=+75.0°;    or  y=+10.7+51.4n, 

where  n  and  m  may  be  any  integer  number. 


L32  ENGINEERING   MA  THEMATICS. 

Since  to  an  angle  jn,  any  multiple  of  2x  or  300  deg.  may 

be  added,  any  multiple  of  :     -  may  be  added  to  the  angle  -A 

and  thus  the  angle  —  may  be  made  positive,  etc. 

91.  The  equation  of  the  temperature  wave  thus  becomes: 
^=8.75-13.69  cos  (0-14.15°) -0.602  cos  2(0-44.95°) 
0.359  cos  3(0 -52.3°) -0.416  cos  4(0-72.95°) 
0.091  cos  5(0- 19.77°) -0.184  cos  6(0-20.5°) 
-0.085  cos  7(0-10.7°);  (a) 

or,  transformed  to  sine  functions  by  the  substitution, 

cos  u>=—  sin  (oj  —  90°): 

7/ =  8.75 +  13.09  sin  (0-104.15°) +0.602  sin  2(0-89.95°) 
+  0.359  sin  3(0-82.3°)  +0.41(5  sin  4(0-95.45°) 
+  0.091  sin  5(0- 109.77°) +0.184  sin  0(0-95.5°) 
+  0.085  sin  7(0-75°).  (&) 

The  cosine  form  is  more  convenient  for  some  purposes, 
die  sine  form  for  other  purposes. 

Substituting  ,9  =  0-14.15°;  or,  5  =  0-104.15°,  these  two 
(••luations  (a)  and  (b)  can  be  transformed  into  the  form, 

^  =  8.75-13.69  cos  ,3-0.62  cos 2(,3-30.8°) -0.359  cos3(.?-3K.15°> 
-0.416  cos  4(,°- 58.8°) -0.091  cos  5(0-5.6°) 

-0.184  cos  603-6.35°) -0.085  cos  7(/?-4S.0°),  (c) 

and 

y  - 8.75+13.69  sin  5+0.602  sin  2(5+ 14.2°) +0.359  sin  3(5+21.85°) 

+0.416  sin  4(5+8.7°)  +0.91  sin  5(5-5.6°) 

+  0.184  sin  6(5+8.65°)  +0.085  sin  7(5  +  29.15°).  (d) 

The  periodic  variation  of  the  temperature  y,  as  expressed 
by  these  equations,  is  a  result  of  the  periodic  variation  of  the 
thermomotive  force;    thai   is,  the  solar  radiation.     This  latter 


TRIGONOMETRIC '  SERIES.  1 33 

is  a  minimum  on  Dec.  22d,  that  is,  9  time-degrees  before  the 
zero  of  0,  hence  may  be  expressed  approximately  by: 

z=c-h  cos  (0+9°); 

or  substituting  /?  respectively  d  for  0: 

z  =  c-h  cos  (/3+23.150) 
=  c+Asin  (£+23.15°). 

This  means:  the  maximum  of  y  occurs  23.15  deg.  after  the 
maximum  of  z;  in  other  words,  the  temperature  lags  23.15  deg., 
or  about  /g  period,  behind  the  thermomotive  force. 

Near  o=0,  all  the  sine  functions  in  (d)  are  increasing;  that 
is.  the  temperature  wave  rises  steeply  in  spring. 

Near  £=180  deg.,  the  sine  functions  of  the  odd  angles  are 
decreasing,  of  the  even  angles  increasing,  and  the  decrease  of 
the  temperature  wave  in  fall  thus  is  smaller  than  the  increase 
in  spring. 

The  fundamental  wave  greatly  preponderates,  with  ampli- 
tude ci  =  13.69. 

In  spring,  for  d  =  —  14.5  deg.,  all  the  higher  harmonics 
rise  in  the  same  direction,  and  give  the  sum  1.74,  or  12.7 
per  cent  of  the  fundamental.  In  fall,  for  d=— 14.5 +7r,  the 
even  harmonics  decrease,  the  odd  harmonics  increase  the 
steepness,  and  give  the  sum   —0.67,  or  —4.9  per  cent. 

Therefore,  in  spring,  the  temperature  rises  12.7  per  cent 
faster,  and  in  autumn  it  falls  4.0  per  cent  slower  than  corre- 
sponds to  a  sine  wave,  and  the  difference  in  the  rate  of  tempera- 
ture rise  in  spring,  and  temperature  fall  in  autumn  thus  is 
12.7+4.9  =  17.6  per  cent. 

The  maximum  rate  of  temperature  rise  is  90—14.5  =  75.5 
deg.  behind  the  temperature  minimum,  and  23.15+75.5  =  98.7 
deg.  behind  the  minimum  of  the  thermomotive  force. 

As  most  periodic  functions  met  by  the  electrical  engineer 
are  symmetrical  alternating  functions,  that  is,  contain  only 
the  odd  harmonics,  in  general  the  work  of  resolution  into  a 
trigonometric  series  is  very  much  less  than  in  above  example. 
Where  such  reduction  lias  to  be  carried  out  frequently,  it  is 
advisable  to  memorize  the  trigonometric  functions,  from  10 
to  10  deg.,  up  to  3  decimals:  that  is,  within  the  accuracy  of 
the  slide  rule,  as  thereby  the  necessity  of  looking  up  tables  is 


134  ENGINEERING  MATHEMATICS. 

eliminated  and  the  work  therefore  done  much  more  expe- 
ditiously. In  general,  the  slide  rule  can  be  used  for  the  calcula- 
tions. 

As  an  example  of  the  simpler  reduction  of  a  symmetrical 
alternating  wave,  the  reader  may  resolve  into  its  harmonics, 
up  to  the  7th,  the  exciting  current  of  the  transformer,  of  which 
the  numerical  values  are  given,  from  10  to  10  deg.  in  Table  X. 

C.  REDUCTION    OF    TRIGONOMETRIC    SERIES    BY  POLY- 
PHASE   RELATION. 

92.  In  some  cases  the  reduction  of  a  general  periodic  func- 
tion, as  a  complex  wave,  into  harmonics  can  be  carried  out 
in  a  much  quicker  manner  by  the  use  of  the  polyphase  equation, 
Chapter  III,  Part  A  (23).  Especially  is  this  true  if  the  com- 
plete equation  of  the  trigonometric  scries,  which  represents  the 
periodic  function,  is  not  required,  but  the  existence  and  the 
amount  of  certain  harmonics  are  to  be  determined,  as  for 
instance  whether  the  periodic  function  contain  even  harmonics 
or  third  harmonics,  and  how  large  they  may  be. 

This  method  does  not  give  the  coefficients  an,  bn  of  the 
individual  harmonics,  but  derives  from  the  numerical  values 
of  the  general  wave  the  numerical  values  of  any  desired 
harmonic.  This  harmonic,  however,  is  given  together  with  all 
its  multiples;  that  is,  when  separating  the  third  harmonic, 
in  it  appears  also  the  6th,  9th,  12th,  etc. 

In  separating  the  even  harmonics  y2  from  the  general 
wave  y,  in  paragraph  84,  by  taking  the  average  of  the  values 
of  y  for  angle  0,  and  the  values  of  y  for  angles  (0+tz),  this 
method  lias  already  been  used. 

Assume  that  to  an  angle   0  there  is  successively  added  a 

constant  quantity  a,  thus:    0;    0  +  a;    0  +  2a;    0  +  3a;    0+4a, 

etc,  until  the  same  angle  0  plus  a  multiple  of  2-  is  reached; 

2m '- 
0  +  na  =  0+2?n7r;    that  is,   a  =  -    -;    or,  in  other  words,   a  is 

1/n  of  a  multiple  of  2-      Then  the  sum  of  the  cosine  as  well 
as  the  sine  functions  of  all  these  angles  is  zero: 

cos  0-Kos  (#+a)+cos  (fl+2a)+cos  (#+3a)+.  .  . 

+cos  (0  +  [n-l]a)=0;       (1) 


TRIGONOMETRIC  SERIES.  135 

sin  0+sin  (0+a)  +sin  (0 +2a)  +sin  (0+3a)  +. . . 

+sm(d+[n-i\a)=0}        (2) 

where 

na  =  2m- (3) 

These  equations  (1)  and  (2)  hold  for  all  values  of  a,  except  for 
a  =  2-,  or  a  multiple  thereof.  For  a  =  2tz  obviously  aJ  the  terms 
of  equation  (1)  or  (2)  become  equal,  and  the  sums  become 
>/  cos  d  respectively  n  sin  0. 

Thus,  if  the  scries  of  numerical  values  of  y  is  divided  into 

2- 
n  successive    sections,    each   covering  —    degrees,    and  these 

i  If 

>eetions  added  together, 

+y(d+[n-l^\        (4) 

In  this  sum,  all  the  harmonics  of  the  wave  y  cancel  by  equations 
(1)  and  (2),  except  the  nth.  harmonic  and  its  multiples, 

an  cos  nO  +  bn  sin  nO;   a2n  cos  2n0+b2n  sin  2n0,  etc. 

in  the  latter  all  the  terms  of  the  sum  (4)  are  equal;  that  is, 
the  sum  (4)  equals  n  times  the  nth  harmonic,  and  its  multiples. 
Therefore,  the  nth  harmonic  of  the  periodic  function  ///together 
with  its  multiples,  is  given  by 

yn(0)=\{y(0)  +y(d  +|)  +  //^+2^  +  ...-f//^+[n-l]^}(5) 

For  instance,  for  n  =  2, 

y2  =  h\y(0)+y(o+-)\, 

gives  the  sum  of  all  the  even  harmonics;  that  is,  gives  the 
second  harmonic  together  with  its  multiples,  the  4th,  6th,  etc., 
as  seen  in  paragraph  7,  and  for,  n  =  3, 

»-l[vw  +•(•+!)  +»(»+t)}' 


136 


ENGINEERING  MA  THEM  A  TI(  'S. 


gives  the  third  harmonic,  together  with  its  multiples,  the  6th, 
9th,  etc. 

This  method  does  not  give  the  mathematical  expression 
of  the  harmonics,  but  their  numerical  values.  Thus,  if  the 
mathematical  expressions  are  required,  each  of  the  component 
harmonics  has  to  be  reduced  from  its  numerical  values  to 
the  mathematical  equation,  and  the  method  then  offers  no 
ad vantage. 

It  is  especially  suitable,  however,  where  certain  classes  of 
harmonics  are  desired,  as  the  third  together  with  its  multiples. 
In  this  case  from  the  numerical  values  the  effective  value, 
that  is,  the  equivalent  sine  wave  may  be  calculated. 

93.  As  illustration  may  be  investigated  the  separation  of 
the  third  harmonics  from  the  exciting  current  of  a  transformer. 

Table  X 


A 

(l) 

0 

(2) 
i 

(3) 
0 

(1) 

i 

(5) 

e 

(6) 
i 

(7) 
n 

0 
10 
20 

30 
40 
50 

60 

+  24.0 
+  20.0 
+  12 

6.5 

-  8.5 

120 
130 
140 

150 
160 

170 

180 

-15.1 
-16.5 
-18.5 

-21 

-  22 . 7 

-  23 . 7 

-24 

240 
250 
260 

271 ) 
280 
290 

300 

+  X.5 
+  10 

+  11 

+  12 
•  13 

+  14 

+  15.1 

+  5.8 
+  4.5 
+  1.5 

1.7 
-3.7 
-5.4 

-  5 . 8 

B 

30 

13 

30 

13 

3d 

u 

is 

0 
30 
60 

1 

+  5.8 
+  4.5 
+  1.5 

120 
150 

ISO 

-3.7 

-5.4 
-5.8 

240 
270 
300 

1  :. 
+  1.7 
+  3.7 

+  0.2 
+  0.3 
-0.2 

In  table  X  A,  are  given,  in  columns  1,  3,  5,  the  angles  6, 
from  10  deg.  to  10  deg.,  and  in  columns  2,  4,  6,  the  correspond- 
ing values  of  the  exciting  current  i,  as  derived  by  calculation 
from  the  hysteresis  cycle  of  the  iron,  or  by  measuring  from  the 


TRIGONOMETRIC  SERIES. 


L3- 


photographic  film  of  the  oscillograph.  Column  7  then  gives 
one-third  the  sum  of  columns  2,  4,  and  6,  that  is,  the  third  har- 
monic with  its  overtones,  £3. 

To  find  the  9th  harmonic  and  ils  overtones  ig,  the  same 
method  is  now  applied  to  is,  for  angle  30.  This  is  recorded 
in  Table  X  B. 

In  Fig.  46  are  plotted  the  total  exciting  current  i,  its  third 
harmonic  13,  and  the  9th  harmonic  ig. 

This  method  has  the  advantage  of  showing  the  limitation 
of  the  exactness  of  the  results  resulting  from  the  limited  num- 


Fig.  46. 


ber  of  numerical  values  of  i,  on  which  the  calculation  is  based. 
Thus,  in  the  example,  Table  X,  in  which  the  values  of  i  are 
given  for  every  10  deg.,  values  of  the  third  harmonic  are  derived 
for  every  30  deg..  and  for  the  9th  harmonic  for  every  90  deg.; 
that  is,  for  the  latter,  only  two  points  per  half  wave  are  deter- 
minable from  the  numerical  data,  and  as  the  two  points  per  half 
wave  are  just  sufficient  to  locate  a  sine  wave,  it  follows  that 
within  the  accuracy  of  the  given  numerical  values  of  i,  the 
9th  harmonic  is  a  sine  wave,  or  in  other  words,  to  determine 
whether  still  higher  harmonics  than  the  9th  exist,  requires  for 
i  more  numerical  values  than  for  every  10  deg. 

As  further  practice,  the  reader  may  separate  from  the  gen- 


i:;s 


ENGINEERING   M.  1  'I'll EM  A  TICS. 


eral  wave  of  current,  io  in  Table  XI,  the  even  harmonics  i2, 
by  above  method, 

and  also  the  sum  of  the  odd  harmonics,  as  the  residue, 

ii=io— %2, 

then  from  the  odd  harmonics  i\  may  be  separated  the  third 
harmonic  and  its  multiples, 

t3  =  i{*i(0)  +ii(0  +  12O  deg.  )+*i (0+240  deg.)}, 

and  in  the  same  manner  from  i3  may  be  separated  its  third 
harmonic;  that  is,  ig. 

Furthermore,  in  the  sum  of  even  harmonics,  from  i2  may 
again  be  separated  its  second  harmonic,  i4,  and  its  multiples, 
and  therefrom,  ig,  and  its  third  harmonic,  i&,  and  its  multiples, 
thus  giving  all  the  harmonics  up  to  the  9th,  with  the  exception 
of  the  5th  and  the  7th.  These  latter  two  would  require  plotting 
the  curve  and  taking  numerical  values  at  different  intervals, 
so  as  to  have  a  number  of  numerical  values  divisible  by  5  or  7. 

It  is  further  recommended  to  resolve  this  unsymmetrical 
exciting  current  of  Table  XI  into  the  trigonometric  series  by 
calculating  the  coefficients  an  and  b„,  up  to  the  7th,  in  the  man- 
ner discussed  in  paragraphs  6  to  8. 

Table  XI 


6 

io 

0 

io 

0 

in 

0 

to 

0 

+  05.7 

90 

-26.7 

ISO 

-34.3 

270 

3.3 

10 

+  7S.7 

100 

-27.3 

190 

-27.3 

2S0 

1  .8 

20 

+  53.7 

110 

-28.1 

200 

-16.8 

290 

+  1.2 

30 

+  23.7 

L20 

-  28 . 8 

210 

11  3 

300 

+  4.7 

10 

-  2.3 

130 

-  29 . 3 

220 

-  8.3 

310 

+  10.7 

50 

16.3 

140 

-  29 . 8 

230 

7.3 

320 

+  22.7 

60 

-22. S 

150 

-31 

'_•!() 

-  6.3 

330 

+  41.7 

70 

2  1 . 3 

L60 

-32.6 

250 

.->  3 

340 

+  65.7 

80 

-25.S 

170 

-  33 .  S. 

260 

1  3 

350 

+  S5.7 

TRIGONOMETRIC  SERIES.  \'A\) 

D.   CALCULATION    OF    TRIGONOMETRIC    SERIES    FROM 
OTHER  TRIGONOMETRIC    SERIES. 

94.  An  hydraulic  generating  station  has  for  a  long  time  been 
supplying  electric  energy  over  moderate  distances,  from  a  num- 
ber of  750-kw.  4400-volt  60-cycle  three-phase  generators.  The 
station  is  to  be  increased  in  size  by  the  installation  of  some 
larger  modern  three-phase  generators,  and  from  this  station 
6000  k\v.  are  to  be  transmitted  over  a  long  distance  Transmis- 
sion line  at  44,000  volts.  The  transmission  line  has  a  length 
of  60  miles,  and  consists  of  three  wires  No.  0  B.  &  S.  with  5 
ft.  between  the  wires. 

The  question  arises,  whether  during  times  of  light  load  the 
old  750-kw.  generators  can  be  used  economically  on  the  trans- 
mission line.  These  old  machines  give  an  electromotive  force 
wave,  which,  like  that  of  most  earlier  machines,  differs  con- 
siderably from  a  sine  wave,  and  it  is  to  be  investigated,  whether, 
due  to  this  wave-shape  distortion,  the  charging  current  of  the 
transmission  line  will  be  so  greatly  increased  over  the  value 
which  it  would  have  with  a  sine  wave  of  voltage,  as  to  make 
the  use  of  these  machines  on  the  transmission  line  uneconom- 
ical or  even  unsafe. 

Oscillograms  of  these  machines,  resolved  into  a  trigonomet- 
ric series,  give  for  the  voltage  between  each  terminal  and  the 
neutral,  or  the  Y  voltage  of  the  three-phase  system,  the  equa- 
tion : 

c  =  e0{sin  0-0.12  sin  (30-2.3°)-O.23  sin  (50-1.5°) 

+0.13  sin  (70-6.2°)}.     .     (1) 

In  first  approximation,  the  line  capacity  may  be  considered 
as  a  condenser  shunted  across  the  middle  of  the  line;  that  is, 
half  the  line  resistance  and  half  the  line  reactance  is  in  series 
with  the  line  capacity. 

As  the  receiving  apparatus  do  not  utilize  the  higher  har- 
monics of  the  generator  wave,  when  using  the  old  generators, 
their  voltage  has  to  be  transformed  up  so  as  to  give  the  first 
harmonic  or  fundamental  of  44,000  volts. 

44.000  volts  between  the  lines  for  delta)  gives  44,000 4-V3  = 
25,400  volts  between  line  and  neutral.     This  is  the  effective 


I  10  ENGINEERING   MATHEMATICS. 

value,  and  the  maximum  value  of  the  fundamental  voltage 
wave  thus  is:  25,400  X  V2  =  36,000  volts,  or  30  kv.;  that  is, 
e0  =  36,  and 

e=36{sin  0-0.12  sin  (30-2.3°)-O.23  sin  (50-1.5°) 

+  0.13  sin  (70-6.2°)},   .     (2) 

would  be  the  voltage  supplied  to  the  transmission  line  at  the 
high  potential  terminals  of  the  step-up  transformers. 

From  the  wire  tables,  the  resistance  per  mile  of  No.  0  B.  &  S. 
copper  line  wire  is  r0  =  0.52  ohm. 

The  inductance  per  mile  of  wire  is  given  by  the  formula : 

L0  =  0.7415  log  £+0.0805mh,      ....     (3) 

where  ls  is  the  distance  between  the  wires,  and  lr  the  radius  of 
the  wire. 

In  the  present  case,  this  gives  ls  =  5  ft.  =  60  in.  lr  =  0 .  1625  in. 
L,)  =  1.9655  mh.,  and,  herefrom  it  follows  that  the  reactance,  at 
/=  60  cycles  is 

x0  =  2-fL0  =  0 . 75  ohms  per  mile (4) 

The  capacity  per  mile  of  wire  is  given  by  the  formula: 

n      °-0408      r 

'  o  =  -    —r-  mf.; (.)) 

log 


bZr 


hence,  in  the  present  case,  Co=0.0159  ml'.,  and  the  condensive 
reactance  is  derived  herefrom  as: 

xro  =  ^—rrr  =  l (')()()()()  ohms;        ....       (i\) 

60  miles  of  line  then  give  the  condensive  reactance, 

x 
.rr  =  ^-J  =  2770  ohms; 

30  miles,  or  half  the  line  (from  the  generating  station  to  the 
middle  of  the  line,  where  the  line  capacity  is  represented  by  a 
shunted  condenser)  give:    the  resistance,  r   =  30r0==  15.6  ohms; 


TRIGONOMETRIC '  SERIES. 


Ill 


the  inductive  reactance,  x  30x0  22.5  ohms,  and  the  equiva- 
lent cireuh  of  the  line  now  consists  of  the  resistance  r,  inductive 
reactance  x  and  condensive  reactance  xc,  in  series  with  each 
other  in  the  circuit  of  the  supply  voltage  e. 

95.  If  i   ■■  current  in  the  line  (charging  current)  the  voltage 
consumed  by  the  line  resistance  r  is  ri. 

di 
The  voltage  consumed  by  the  inductive  reactance  x  is  .r -7- • 

J  (Hi 

the  voltage  consumed  by  the  condensive  reactance  xr  is  xr  I  idd, 
and  therefore, 


di 


e  =  x-Ta+ri+xc  J  idd. 


•/• 


(7) 


Differentiating  this  equation,  for  the  purpose  of  eliminating 
the  integral,  gives 


de       d2i       di 


dO    xd62 


rdd+Xct> 


or 


(S) 


de 

To 


22.5 


dH     ,  .  ,(/-'      ,__,. 


The  voltage  e  is  given  by  (2),  which  equation,  by  resolving 
the  trigonometric  functions,  gives 

e     36  sin  0-4.32  sin  30-8.28  sin  50  +4  .64  sin  70 

+0.18  cos  30 +0.22  cos  50-0. 50  cos  70;    .     (0) 


hence,  differentiating, 
de 


dll 


36  cos  0-12.96  cos  30-41.4  cos  50+32.5  cos  70 
-0.54  sin  30     11  sin  50+3.5  sin  70. 


ao) 


Assuming  now  for  the  current   /'  a  tiigonometric  series  with 
indeterminate  coefficients, 


i  =  (l\  cos  I)  I  CL%  cos  30   I  ";,  cos  50   l  "7  cos  70 

-+  b]  sin  0  '  &3  sin  30  ■!  b5  sin  50+b7  sin  70, 


(111 


142 


ENGINEERING  MA T  HEMATICS. 


• 


(12) 


substitution  of  (10)  and  (11)  into  equation  (8)  must  give  an 
identity,  from  which  equations  for  the  determination  of  an  and 
6„  are  derived;  that  is,  since  the  product  of  substitution  must 
be  an  identity,  all  the  factors  of  cos  0,  sin  0,  cos  30,  sin  30, 
etc.,  must  vanish,  and  this  gives  the  eight  equations: 

30       =2770ai+  15.06,-  22.5ai;l 

0         =27706!-  15.0a!-  22.5&i; 

-12.96=2770a3  +  4'6.858-  202. 5a3; 

-  0.54= 277063-  46.8a3-  202. 563; 
-41.4  =2770a5  +  7865-  502. 5a5; 

-  1.1  =  27706s-   78a5-  50.2565; 
32 . 5  =  2770a7  + 109 .  267  - 1 102 .  5a7 ; 

3.5  =  277067- 109. 2a7- 1102. 567.  . 

Resolved,  these  equations  give 

oi=     13.12; 

6i=  0.07 
a3=-  5.03: 
63  =  -  0.30 
a5= -18.72 
65=-  1.15 
a7=     19.30; 

67=       3.37; 
hence, 

i  =  13 .  12  cos  0-5.03  cos  30-18.72  cos  50  +  19.30  cos  70 
+0.07  sin 0-0.30  sin  30-1.15  sin  50+3.37  sin  70 
=  13.12  cos  (0-O.3°)-5.O4cos  (30-3.3°) 
- 18 .  76  cos  (50 -  3 . 0°)  +  19. 59  cos  (70 -  9 . 9°). 


(13) 


(14) 


TRIGONOMETRIC  SERIES. 


143 


96.  The  effective  value  of  this  current  is  given  as  the  square 
root  of  the  sum  of  squares  of  the  effective  values  of  the  indi- 
vidual harmonics,  thus: 


/  =  ^^|~+]T^21.l>amp. 


As  the  voltage  between  line  and  neutral  is  25,400  effective, 
this  gives  Q= 25,400X21. 6  =  540,000  volt-amperes,  or  540  kv.- 
amp.  per  line,  thus  a  total  of  3Q  =  1620  kv.-amp.  charging  cur- 
rent of  the  transmission  line,  when  using  the  e.m.f.  wave  of 
these  old  generators. 

It  thus  would  require  a  minimum  of  3  of  the  750-kw. 
generators  to  keep  the  voltage  on  the  line,  even  if  no  power 
whatever  is  delivered  from  the  line. 

If  the  supply  voltage  of  the  transmission  line  were  a  perfect 
sine  wave,  it  would,  at  44,000  volts  between  the  lines,  be  given 
by 

ex  =30  sin  0, (15) 

which  is  the  fundamental,  or  first  harmonic,  of  equation  (9). 

Then  the  current  i  would  also  be  a  sine  wave,  and  would  be 
given  by 

i'i  =rti  cos  0  +hi  sin  0, 


13.12  cos  6  +0.07  sin  0,    \, 

13.12  cos  (0-0.3°), 


.     •     (16) 


and  its  effective  value  would  be 


13.12 
/i=-         =9.3  amp. 


(17) 


This  would  correspond  to  a  kv.-amp.  input  to  the  line 

3Qi  =  3x25.4x9.3  =  710  kv.-amp. 

The  distortion  of  the  voltage  wave,  as  given  by  equation  (1), 
thus  increases  the  charging  volt-amperes  of  the  line  from  710 


144 


EXdIXEERING  MA  THEM  A  TICS. 


kv.-amp.  to  1620  kv.-amp.  or  2.28  times,  and  while  with  a  sine 
wave  of  voltage,  one  of  the  750-kw.  generators  would  easily  be 
able  to  supply  the  charging  current  of  the  line,  due  to  the 


Fig.  47. 


wave  shape  distortion,  more  than  two  generators  are  required. 
It  would,  therefore,  not  be  economical  to  use  these  generators 
on  the  transmission  line,  if  they  can  be  used  for  any  other 
purposes,  as  short-distance  distribution. 


Fig.  is. 


In  Figs.    17  and  48  are  plot  led  the  voltage  wave  and  the 
currenl   wave,  from  equations  (9)  and  (14)   respectively,  and 


TRIGONOMETRIC  SERIES.  L45 

the  numerical  values,  from    10  cleg,   to    10  deg.,  recorded   in 
Table  XII. 

In  Figs.  47  and  48  the  fundamental  sine  wave  of  voltage 
and  current  are  also  shown.  As  seen,  the  distortion  of  current 
is  enormous,  and  the  higher  harmonies  predominate  over  the 
fundamental.  Such  waves  are  occasionally  observed  as  charg- 
ing currents  of  transmission  lines  or  cable  systems. 

97.  Assuming  now  that  a  reactive  coil  is  inserted  in  series 
with  the  transmission  line,  between  the  step-up  transformers 
and  the  line,  what  will  be  the  voltage  at  the  terminals  of  this 
reactive  coil,  with  the  distorted  wave  of  charging  current 
traversing  the  reactive  coil,  and  how  does  it  compare  with  the 
voltage  existing  with  a  sine  wave  of  charging  current? 

Let  L  =  inductance,  thus  x  =  2/r/L  =  reactance  of  the  coil; 
and  neglecting  its  resistance,  the  voltage  at  the  terminals  of 
the  reactive  coil  is  given  by 

'—*?» (18) 

Substituting  herein  the  equation  of  current,  (11),  gives 

e'  =  x\a\  sin  0+3a,3  sin  30+5as  sin  50 +7a7  sin  76  } 

.      (19) 
- 61  cos  0-3&3  cos ?j0-5b5  cos  50-7b7  cos  70 > ; 

hence,  substituting  the  numerical  values  (13), 

e'=zj  13.12  sin  0-15.09  sin  3(9-93.6  sin  50+135.1  sin  76  1 

-0.07  cos  0  +0.00  cos  30+5.75  cos  50 -23.0  cos  7// 1 

=  x\  13.12  sin  (0-0.3°) -15.12  sin  (30-3.3°) 

-93.8  sin  (50-3.0°)  +139.1  sin  (70-9.9°) }. 

This  voltage  gives  the  effective  value 


(20) 


E'=xV\\  13.122  +  15.122+93.82  +  139.12|  =  119.4.r, 

while  the  effective  value  with  a  sine  wave  would  be  from  (17), 

El>  =  xIl=C).:\.r: 

hence,  the   voltage   across  the   reactance  x  has   been   increased 
12.8  times  by  the  wave  distortion. 


146 


ENGINEERING   MATHEMATK  'S. 


The  instantaneous  values  of  the  voltage  e'  are  given  in  the 
last  column  of  Table  XII,  and  plotted  in  Fig.  49,  for  x  =  l. 
As  seen  from  Fig.  49,  the  fundamental  wave  has  practically 


Fig.  49. 


vanished,  and  the  voltage  wave  is  the  seventh  harmonic,  modi- 
fied bv  the  fifth  harmonic. 


Table  XII 

0 

e 

£ 

.■' 

0 

e 

i 

e' 

0 
10 
20 

-0.10 

+  2.23 

3.74 

+   8.67 

+    5.30 
0.86 

17 

+   46 
+     3 

90 
100 

110 

27.41 

31.77 
40.57 

4.15 
•  26.19 

+  24.99 

-200 
106 

+  119 

30 
40 
50 

7.17 
17.35 
31.70 

+    7.39 
+  30.39 
+  38.58 

+  131 
-116 

+   36 

120 
130 
140 

42.70 
33.14 
IS.03 

8.10 
38.79 

-36.65 

+  182 

+    93 
96 

60 
80 

12.06 
40.33 
32.87 

+  15.66 
-19.01 
-29.13 

+  167 

+  159 

51 

150 
160 

170 

6.99 
2.ss 

1    97 

-13.41 

+    2.43 
1.00 

138 

31 
+    54 

90 

27 .  1 1 

4.15 

200 

ISO 

+  0.10 

8.67 

+    17 

,__ 

CHAPTER  IV. 


MAXIMA    AND    MINIMA. 


98.  In  engineering  investigations  the  problem  of  determin- 
ing the  maxima  and  the  minima,  that  is,  the  extrema  of  a 
function,  frequently  occurs.  For  instance,  the  output  of  an 
electric  machine  is  to  be  found,  at  which  its  efficient'}'  is  a  max- 
imum, or,  it  is  desired  to  determine  that  load  on  an  induction 
motor  which  gives  the  highest  power-factor;    or,  that  voltage 


Y 

A 

1 

3' 'If 

X" 

p 

P 

?J\ 

-Q 

p 

u 

p3 

> 

0 

/ 

X 

Fig.  50.     Graphic  Solution  of  Maxima  and  Minima. 

which  makes  the  cost  of  a  transmission  line  a  minimum;  or, 
that  speed  of  a  steam  turbine  which  gives  the  lowest  specific 
steam  consumption,  etc. 

The  maxima  and  minima  of  a  function,  y=f(x),  can  be  found 
by  plotting  the  function  as  a  curve  and  taking  from  the  curve 
the  values  x,  y,  which  give  a  maximum  or  a  minimum.  For 
instance,  in  the  curve  Fig.  50,  maxima  are  at  Pi  and  P2,  minima 
at  P3  and  P4.  This  method  of  determining  the  extrema  of 
functions  is  necessary,  if  the  mathematical  expression  between 

147 


I  18 


ENGINEERING  MAT 11  EM  A  TICS. 


x  and  ?/,  that  is,  the  function  y=f(x),  is  unknown,  or  if  the 
function  y=f(x)  is  so  complicated,  as  to  make  the  mathematical 
calculation  of  the  extrema  impracticable.  As  examples  of 
this  method  the  following  may  be  chosen: 


ie 

— — 

u 

— 10- 

— s 

c 

6 

0 

X 

J       i 

' 

?       > 

1       1 

)      1 

2 

1 

i      1 

3      1 

i      2 

) 

:.' 

2 

21 

26 

1 

2* 

30 

Fig.  51.     Magnetization  Curve. 

Example  i.  Determine  that  magnetic  density  03,  at  which 
the  permeability  [x  of  a  sample  of  iron  is  a  maximum.  The 
relation  between  magnetic  field  intensity  3C,  magnetic  density 
03  and  permeability  \i  cannot  be  expressed  in  a  mathematical 
equation,  and  is  therefore    usually  given  in  the   form  of  an 


1400 

Xs* 

1 

/'mc 

-800- 

— V 

/UK) 

SB 

] 

• 

> 

1 

. 

>     < 

> 

■ 

i 

)     i 

i)      l 

Kilo-lines 
1      12      13 

l 

i      l 

5 

Fig.  52.     Permeability  Curve. 

empirical  curve,  relating  (B  and  3C,  as  shown  in  Fig.  ,51.     From 
this  curve,  corresponding  values  of  (B  and  3C  are  taken,  and  their 

ratio,  that  is,  the  permeability  j«=— ,  plotted  against®  as  abscissa. 
This  is  done  m  Fig.  52.     Fig.  52  then  shows  that  a  maximum 


MAXIMA    AND  MINIMA. 


L49 


occurs  at  point  /<m;lx,  for  (B  =  10.2  kilolines,  /t  =  1340,  and  minima 
at  the  starting-point  P2,  for  (B=0,  ^=370,  and  also  for  (ft  =  oo, 
where  by  extrapolation  fi=l. 

Example  2.  Find  that  output  of  an  induction  motor 
which  gives  the  highest  power-factor.  While  theoretically 
an  equation  can  be  found  relating  output  and  power-factor 
of  an  induction  motor,  the  equation  is  too  complicated  for  use. 
The  most  convenient  way  of  calculating  induction  motors  is 
to  calculate  in  tabular  form  for  different  values  of  slip  s,  the 
torque,  output,  current,  power  and  volt -ampere  input,  efficiency, 
power-factor,  etc.,  as  is  explained  in  "Theoretical  Elements 
of  Electrical  Engineering,"  third  edition,  p.  303.     From  this 


Cos0/ 
0.90 

0.88 
0.86 

0.84.: 

0.82 


Fig.  53.     Power-factor  Maximum  of  Induction  Motor. 


R 

p 
0 

P2 

20 

BO 

30 

30 

41 

P 

on 

50 

B0 

00 

30    V 

/atts 

table  corresponding  values  of  power  output  P  and  power- 
factor  cos  0  are  taken  and  plotted  in  a  curve,  Fig.  53,  and  the 
maximum  derived  from  this   curve   is  P  =  4120,  cos  0  =  0.904. 

For  the  purpose  of  determining  the  maximum,  obviously 
not  the  entire  curve  needs  to  be  calculated,  but  only  a  short 
range  near  the  maximum.  This  is  located  by  trial.  Thus 
in  the  present  instance,  P  and  cos  0  are  calculated  for  s  =  0.1 
and  s  =  0.2.  As  the  latter  gives  lower  power-factor,  the  maximum 
power-factor  is  below  *  =  0.2.  Then  s  =  0.05  is  calculated  and  gives 
a  higher  value  of  cos  I)  than  .$  =  0.1;  that  is,  the  maximum  is 
below  *  0.1.  Then  s  =  0.02  is  calculated,  and  gives  a  lower 
value  of  cos  6  than  s  =  0.05.  The  maximum  value  of  cos  0 
thus  lies  between  s  =  0.02  and  .$  =  0.1,  and  only  the  part  of  the 
curve  between  s  =  0.02  and  s  =  0.1  needs  to  be  calculated  for 
the  determination  of  the  maximum  of  cos  d,  as  is  done  in  Fig.  53. 

99.  When  determining  an  extremum  of  a  function  ;/=f(x). 
by  plotting  it  as  a  curve,  the  value  of  x,  at  which  the  extreme 


1  ;>< )  ENGINEERING   M .  1  THEM  A  TICS. 

occurs,  is  more  or  less  inaccurate,  since  at  the  extreme  the 
curve  is  horizontal.  For  instance,  in  Fig.  53,  the  maximum 
of  the  curve  is  so  fiat  that  the  value  of  power  P,  for  which 
cos  6  became  a  maximum,  may  be  anywhere  between  P  =  4000 
and  P  =  4300,  within  the  accuracy  of  the  curve. 

In  such  a  case,  a  higher  accuracy  can  frequently  be  reached 
by  not  attempting  to  locate  the  exact  extreme,  but  two  points 
of  the  same  ordinate,  on  each  side  of  the  extreme.  Thus  in 
Fig.  53  the  power  P0,  at  which  the  maximum  power  factor 
cos  0  =  0.904  is  reached,  is  somewhat  uncertain.  The  value  of 
power-factor,  somewhat  below  the  maximum,  cos  0  =  0.90, 
is  reached  before  the  maximum,  at  Pi  =3400,  and  after  the 
maximum,  at  P2  =  4840.  The  maximum  then  may  be  calculated 
as  half-way  between  Pi  and  P2,  that  is,  at  P0  =  ||Pi+P2}  = 
4120  watts. 

This  method  gives  usually  more  accurate  results,  but  is 
based  on  the  assumption  that  the  curve  is  symmetrical  on 
both  sides  of  the  extreme,  that  is,  falls  off  from  the  extreme 
value  at  the  same  rate  for  lower  as  for  higher  values  of  the 
abscissas.  Where  this  is  not  the  case,  this  method  of  inter- 
polation docs  not  give  the  exact  maximum. 

Example  3.  The  efficiency  of  a  steam  turbine  nozzle, 
that  is,  the  ratio  of  the  kinetic  energy  of  the  steam  jet  to  the 
energy  of  the  steam  available  between  the  two  pressures  between 
which  the  nozzle  operates,  is  given  in  Fig.  54,  as  determined  by 
experiment.  As  abscissas  are  used  the  nozzle  mouth  opening, 
that  is,  the  widest  part  of  the  nozzle  at  the  exhaust  end,  as 
fraction  of  that  corresponding  to  the  exhaust  pressure,  while 
the  nozzle  throat,  that  is,  the  narrowest  part  of  the  nozzle,  is 
assumed  as  constant.  As  ordinates  are  plotted  the  efficiencies. 
This  curve  is  not  symmetrical,  but  falls  off  from  the  maximum, 
on  the  sides  of  larger  nozzle  mouth,  far  more  rapidly  than  on 
the  side  of  smaller  nozzle  mouth.  The  reason  is  that  with 
too  large  a  nozzle  mouth  the  expansion  in  the  nozzle  is  carried 
below  the  exhaust  pressure  p>,  and  steam  eddies  are  produced 
by  this  overexpansion. 

The  maximum  efficiency  of  94.6  per  cent  is  found  at  the  point 
/'0,  at  which  the  nozzle  mouth  corresponds  to  the  exhaust 
pressure.  If,  however,  the  maximum  is  determined  as  mid- 
way between  two  points  I\  and  P2,  on  each  side  of  the.  maximum, 


MAXIMA    AND  MIX  IMA. 


151 


at  which  the  efficiency  is  the  same,  9^  per  cent,  a  point  Pq  is 
obtained,  which  lies  on  one  side  of  the  maximum. 

With  unsymmetrical  curves,  the  method  of  interpolation 
thus  docs  not  give  the  exact  extreme.  For  most  engineering 
purposes  this  is  rather  an  advantage.  The  purpose  of  deter- 
mining the  extreme  usually  is  to  select  the  most  favorable 
operating  conditions.  Since,  however,  in  practice  the  operating 
conditions  never  remain  perfectly  constant,  but  vary  to  some 
extent,  the  most  favorable  operating  condition  in  Fig.  54  is 
not  that  where  the  average  value  gives  the  maximum  efficiency 


«J6 — 

SL 

P° 

yi 

^ 

VP- 

J4 

90-c- 

o 
88  u, 

Q. 

o 

{j5J-"J 

80 

78 

0 

0 

{) 

1 

0 

No 

s 

zzle 
0 

bpe 
.9 

ung 
1 

0 

1 

,1 

1 

2 

l.:i 

Fig.  54.     Steam  Turbine  Nozzle  Efficiency;  Determination  of  Maximum. 

(point  Pq),  but  the  most  favorable  operating  condition  is  that, 
where  the  average  efficiency  during  the  range  of  pressure,  occurr- 
ing in  operation,  is  a  maximum. 

If  the  steam  pressure,  and  thereby  the  required  expansion 
ratio,  that  is,  the  theoretically  correct  size  of  nozzle  mouth, 
should  vary  during  operation  by  25  per  cent  from  the  average, 
when  choosing  the  maximum  efficiency  point  Po  as  average, 
the  efficiency  during  operation  varies  on  the  part  of  the  curve 
between  Pi  (91.4  per  cent)  and  P->  (85.2  per  cent),  thus  averaging 
lower  than  by  choosing  the  point  Po'(6.25  per  cent  below  Po) 
as  average.  In  the  latter  case,  the  efficiency  varies  on  the 
part  of  the  curve  from  the  Pi'(90.1  per  cent)  to  P2,(90.1  per 
cent).     (Fig.  55.) 


i:>j 


ENGINEERING   MA  THEM .  1  TICS. 


Thus  in  apparatus  design,  when  determining  extrema  of 
a  function  y=f(x),  to  select  them  as  operating  condition, 
consideration  must  be  given  to  the  shape  of  the  curve,  and 
where  the  curve  is  unsymmetrical,  the  most  efficient  operating 
point  may  not  lie  at  the  extreme,  but  on  that  side  of  it  at  which 
the  curve  falls  off  slower,  the  more  so  the  greater  the  range  of 
variation  is,  which  may  occur  during  operation.  This  is  not 
always  realized. 

ioo.  If  the  function  y=f(x)  is  plotted  as  a  curve,  Fig. 
50,  at  the  extremes  of  the  function,  the  points  1\,  P>,  P;>,,  Pi 
of  curve  Fig.  50,  the  tangent  on  the  curve  is  horizontal,  since 


«J— 

P' 

1  n 

£^ 

3 
1^- 

\r4 

■*-* 

Pi' 

i)r 

88  <x> 

Q_ 

^P= 

SI) 

, 

0 

6 

0 

.7 

0 

No 

8 

zzle 
0 

Dper 
9 

ing 
1 

.0 

1 

.1 

1 

.2 

La 

FlG.  ">•").     Steam  Turbine  -Nozzle  Efficiency;   Determination  of  Maximum. 


at  the  extreme  the  function  changes  from  rising  to  decreasing 
(maximum,  l\  and  P2),  or  from  decreasing  to  increasing  (min- 
imum, P3  and  P4),  and  therefore  for  a  moment  passes  through 
the  horizontal  direction. 

In  general,  the  tangent  of.  a  curve,  as  that  in  Fig.  50,  is  the 
line  which  connects  two  points  P'  and  P"  of  the  curve,  which 
are  infinitely  close  together,  and,  as  seen  in  Fig.  50,  the  angle 
0,  which  this  tangent  PT"  makes  with  the  horizontal  or  X-axis, 


thus  is  given  l>\ 


tan  0  = 


P"Q    dy 
P'Q  ~dx 


MAXIMA    AND  MINIMA.  153 

At    the  extreme,  the  tangent  on  the  curve  is  horizontal, 

that  is,  4-#  =  0,  and,  therefore,  it  follows  that  at  an  extreme 
of  the  function, 

■      y=f{x), (l) 

f-0 (2) 

ax 

The  reverse,  however,  is  not  necessarily  the  case;    that  is, 

.,  .  dy  ...  . 

it  at  a  point  x,  y  :  j-— 0,  this  point  may  not   be  an  extreme; 

that  is,  a  maximum  or  minimum,  but  may  be  a  horizontal 
inflection  point,  as  points  P5  and  J\,  are  in  Fig.  50. 

With  increasing  x,  when  passing  a  maximum  (Pi  and  P?, 
Fig.  50),  y  rises,  then  stops  rising,  and  then  decreases  again. 
When  passing  a  minimum  (P3  and  P4)  y  decreases,  then  stops 
decreasing,  and  then  increases  again.  When  passing  a  horizontal 
inflection  point,  y  rises,  then  stops  rising,  and  then  starts  rising 
again,  at  P5,  or  y  decreases,  then  stops  decreasing,  but  then 
starts  decreasing  again  (at  P6). 

The  points  of  the  function  y=f(x),  determined  by  the  con- 
dition, -t-=0,  thus  require  further  investigation,  whether  they 

represent  a  maximum,  or  a  minimum,  or  merely  a  horizontal 
inflection  point. 

This  can  be  done  mathematically:  for  increasing  x,  when 
passing  a  maximum,  tan  0  changes  from  positive  to  negative; 

that  is,  decreases,   or  in   other  words,  -7-  (tan  0)<Q.     Since 

ax 

tan  0  =-r-,  it  thus  follows  that  at  a  maximum  -j-^  <  0.    Inversely, 

ax  ax- 

at  a  minimum  tan  0  changes  from  negative  to  positive,  hence 
increases,  that  is,  -,-  (tan  0)  >  0;  or,  -j—2>  0.     When   passing 

a  horizontal    inflection   point  tan  0  first    decreases  to  zero  at 

the  inflection  point,  and  then  increases  again;    or,  inversely, 

tan  0  first  increases,  and  then  decreases  again,  that  is,  tan  0  = 
j 

-r  has  a  maximum  or  a  minimum  at  the  inflection  point,  and 
ax 

therefore,  -y~.  ftan  U)="T7>  =  ^  a^  the  inflection  point. 


154  ENGINEERING   MATHEMATICS. 

In  engineering  problems    the   investigation,   whether  the 

dy 

solution   of   the   condition   of   extremes,   t.  =  0,   represents   a 

minimum,  or  a  maximum,  or  an  inflection  point,  is  rarely 
required,  but  it  is  almost  always  obvious  from  the  nature  of 
the  problem  whether  a  maximum  of  a  minimum  occurs,  or 
neither. 

For  instance,  if  the  problem  is  to  determine  the  speed  at 
which  the  efficiency  of  a  motor  is  a  maximum,  the  solution 
speed  =  0,  obviously  is  not  a  maximum  but  a  mimimum,  as  at 
zero  speed  the  efficiency  is  zero.  If  the  problem  is,  to  find 
the  current  at  which  the  output  of  an  alternator  is  a  maximum, 
the  solution  i  =  0  obviously  is  a  minimum,  and  of  the  other 
two  solutions.  ix  and  i2,  the  larger  value,  i2,  again  gives  a 
minimum,  zero  output  at  short-circuit  current,  while  the  inter- 
mediate value  i\  gives  the  maximum. 

101.  The  extremes  of  a  function,  therefore,  are  determined 
by  equating  its  differential  quotient  to  zero,  as  is  illustrated 
by  the  following  examples: 

Example  4.  In  an  impulse  turbine,  the  speed  of  the  jet 
(steam  jet  or  water  jet)  is  S\.  At  what  peripheral  speed  S2  is 
the  output  a  maximum. 

The  impulse  force  is  proportional  to  the  relative  speed  of 
the  jet  and  the  rotating  impulse  wheel;  that  is,  to  {S1-S2). 
The  power  is  impulse  force  times  speed  S2;    hence, 

P  =  kS2(Si-S2), (3) 

1 1  > 
and  is  an  extreme  for  the  value  of  S2,  given  l>v  -jcr     ():    hence. 

Si-2£2  =  0     and     S2=^; (4) 

that  is,  when  the  peripheral  speed  of  the  impulse  wheel  equals 
half  the  jet  velocity. 

Example  5.  In  a  transformer  of  constant  impressed 
e.in.f.  c()  2300  volts;  the  constant  loss,  that  is,  loss  which 
is  independent  of  the  output  (iron  loss),  is  Pi  500  watts.  The 
internal  resistance  (primary  and  secondary  combined)  is  r=20 


MAXIMA    AND  MINIMA.  L55 

ohms.     At  what  currenl   i  is  the  efficiency  of  the  transformer 
a  maximum;    that  is  the  percentage  loss,  /,  a  minimum? 

The  loss  is  P  =  Pi+ri2  =  500  +20i2 (5) 

The  power  input  is  Pi  =ci  =  2300i;  .     .     .     .  (6) 

hence,  the  percentage  loss  is, 

,      P      Pi+ri2  m 

X=p-r^^> (7) 

and  this  is  an  extreme  for  the  value  of  current  i,  given  by 

dX 


\~\  f\  VI  /I  /% 

-t-.=0; 

nence, 

(Pi  +  ri2)e—ei(2ri)      A 

v'i- 

or, 

Pi- 

Ipi 

-  /•/-'  =  0     and     ? '  =  \/~  =  5  amperes 

•     •     (8) 

and  the  output  is  P0  =  ei=ll,500  watts.  The  loss  is,  P  =  P{  + 
ri2  =  2Pi  =  1000  watts;  that  is,  the  i2r  loss  or  variable  loss,  is 
equal  to  the  constant  loss  P{.     The  percentage,  loss  is, 

P     \  7v 
/=p-=  — ■ — =0.087=8.7  percent, 

and  the  maximum  efficiency  thus  is, 

1-,*  =  0.913  =  91 .3  per  cent. 

102.  Usually,  when  the  problem  is  given,  to  determine 
those  values  of  x  for  which  y  is  an  extreme,  y  cannot  be  expressed 
directly  as  function  of  x,  y=f(x),  as  was  done  in  examples 
(4)  and  (5),  but  y  is  expressed  as  function  of  some  other  quan- 
ties,  y=f(u,  /'..),  and  then  equations  between  u,  v..  and  x 
are  found  from  the  conditions  of  the  problem,  by  which  expres- 
sions of  x  are  substituted  for  u,  v  .  .,  as  shown  in  the  following 
example : 

Example  6.  There  is  a  constant  current  i0  through  a  cir- 
cuit  containing  a   resistor   of   resistance   r().     This   resistor   r<j 


1 51  i  ENGINEERING   MA  111  EM  A  TICS. 

is  shunted  by  a  resistor  of  resistance  r.  What  must  be  the 
resistance  of  this  shunting  resistor  r,  to  make  the  power  con- 
sumed in  r,  a  maximum?     (Fig.  o<>.) 

Let  i  be  the  current  in  the  shunting  resistor  r.  The  power 
consumed  in  r  then  is, 

P=ri2 (9) 

The  current  in  the  resistor  r0  is  i0— i,  and  therefore  the 
voltage  consumed  by  r0  is  r0(i0-i),  and  the  voltage  consumed 
by  r  is  ri,  and  as  these  two  voltages  must  be  equal,  since  both 


AAA 


Fig.  50.     Shunted  Resistor. 

resistors  are  in  shunt  with  each  other,  thus  receive  the  same 
voltage, 

ri=r0(i0— i)} 

and,  herefrom,  it  follows  that, 

i=-^-io (10) 

Substituting  this  in  equation  (9)  gives, 

P=  rr°2il)2  (11) 

and  this  power  is  an  extreme  for  "tt  =  ();  hence: 

(r  +  r0)2ro2i02  -  rr0HJ2(r  +  r0) 
(r  +  r0)4 
hence, 

r  =  r0; (12) 

that  is,  the  power  consumed  in  r  is  a  maximum,  if  the  resistor 
r  of  the  shunt  equals  the  resistance  r{). 


MAXIMA    AND  MINIMA.  157 

The.  current  in  r  then  is,  by  equation  (10), 


and  the  power  is, 


pjrtfi(? 


103.  If,  after  the  function  y=f(x)  (the  equation  (11)  in 
example    (6)  )  has   been  derived,  the  differentiation   y;  =  0    is 

immediately  carried  out,  the  calculation  is  very  frequently 
much  more  complicated  than  necessary.  It  is,  therefore, 
advisable  not  to  differentiate  immediately,  but  first  to  simplify 
the  function  y=f(x). 

If  y  is  an  extreme,  any  expression  differing  thereform  by 
a  constant  term,  or  a  constant  factor,  etc.,  also  is  an  extreme. 
So  also  is  the  reciprocal  of  y,  or  its  square,  or  square  root,  etc. 

Thus,  before  differentiation,  constant  terms  and  constant 
factors  can  be  dropped,  fractions  inverted,  the  expression 
raised  to  any  power  or  any  root  thereof  taken,  etc. 

For  instance,  in  the  preceding  example,  in  equation  (11), 

o  ■   •> 

_     7T(T  /  0" 


(r  +  r()r 


the  value  of  r  is  to  be  found,  which  makes  P  a  maximum. 
If  P  is  an  extreme, 

r 
?yi=(r+r„)-" 

which  differs  irom  P  by  the  omission  of  the  constant  factor 

r02i"o2,  also  is  an  extreme. 

The  reverse  of  y  1 , 

(r  +  r0)2 
V-2  =  — - — , 

is  also  an  extreme.     {1/2  is  a  minimum,  where  y\  is  a  maximum, 
and  inversely.) 

Therefore,  the  equation  (11)  can  be  simplified  to  the  form: 

V2  =  --y—  =  r+2r0+—, 


l.')S  ENGINEERING   MATHEMATICS. 

and,  leaving  out  the  constant  term  2r0,  gives  the  final  form. 

y*=r+r-T (13) 


r 
This  differentiated  gives, 

<fya_1   Jo2 
dr  r2  ~U; 

hence, 

r  =  /•(,. 

104.  Example  7.  From  a  source  of  constant  alternating 
e.m.f.  c,  power  is  transmitted  over  a  line  of  resistance  tq  and 
reactance  x0  into  a  non-inductive  load.  What  must  be  the 
resistance  r  of  this  load  to  give  maximum  power? 

If  i  =  current  transmitted  over  the  line,  the  power  delivered 
at  the  load  of  resistance  r  is 

P  =  ri2 (14) 

The  total  resistance  of  the  circuit  is  r  +  r0;  the  reactance 
is  Xq;    hence  the  current  is 

\/(.r+r0)2+.ro2 
and,  by  substituting  in  equation  (14),  the  power  is 

o 

P  =  (r  +  r0)2+x7 (16) 

if  P  is  an  extreme,  by  omitting  e2  and  inverting, 

(r  +  r0)2+x02 
2/i=- " 

=r+2r0+ — - — ■, 

is  also  an  extreme,  and  likewise, 

,  roM-.ro2 

is  an  extreme. 


MAXIMA    AND  MIX  IMA.  159 

Differentiating,  gives: 

dy2     ,      r02+.r02 

-,—  =  1  —  7, =  U, 

dr  r- 


r=\/r02+Xo2 (17) 

Wherefrom  follows,  by  substituting  in  equation  (1G), 


P 


\ 

V 

:  +  .T0^2 

(r0  +  v 

+  x02)2  +  . 

Co2 

•> 

C" 

2(r0  +  \  r02+z02) 


(18) 


Very  often  the  function  y=f(x)  can  by  such  algebraic 
operations,  which  do  not  change  an  extreme,  be  simplified  to 
such  an  extent  that  differentiation  becomes  entirely  unnecessary, 
but  the  extreme  is  immediately  seen;  the  following  example 
will  serve  to  illustrate: 

Example  8.  In  the  same  transmission  circuit  as  in  example 
(7),  for  what  value  of  r  is  the  current  i  a  maximum? 

The  current  i  is  given,  by  equation  (15), 


%=■ 


V{r  +  r0)2+x02 
Dropping  e  and  reversing,  gives, 


.Vi  =  V/(r  +  r0)2+x02; 
Squaring,  gives, 

.'/_-  =  (r  +  r0)2+T02; 

dropping  the  constant  term  xq2  gives 

//3=(r  +  r())2;  (19) 

taking  the  square  rout  gives 

i/4=r+r0; 


1  ( i( )  ENGINEERING  M  A  THEM  A  TICS. 

dropping  the  constant  term  r0  gives 

ys  =  r- (20) 

that  is,  the  current  i  is  an  extreme,  when  y5=r  is  an  extreme, 
and  this  is  the  case  for  r=0  and  r=  go  :   r  =  0  gives, 

(21) 


^  =  - 


as  the  maximum  value  of  the  current,  and  r  =  co  gives 

i  =  (), 

as  the  minimum  value  of  the  current. 

With  some  practice,  from  the  original  equation  (1),  imme- 
diately, or  in  very  few  steps,  the  simplified  final  equation  can 
be  derived. 

105.  In  the  calculation  of  maxima  and  minima  of  engineer- 
ing quantities  x,  y,  by  differentiation  of  the  function  y=f(x), 
it  must  be  kept  in  mind  that  this  method  gives  the  values  of 
x,  for  which  the  quantity  y  of  the  mathematical  equation  y  =f(x) 
becomes  an  extreme,  but  whether  this  extreme  has  a  physical 
meaning  in  engineering  or  not  requires  further  investigation; 
that  is,  the  range  of  numerical  values  of  x  and  y  is  unlimited 
in  the  mathematical  equation,  but  may  be  limited  in  its  engineer- 
ing application.  For  instance1,  if  x  is  a  resistance,  and  the 
differentiation  of  y=f(x)  leads  to  negative  values  of  .r,  these 
have  no  engineering  meaning;  or,  if  the  differentiation  leads 
to  values  of  x,  which,  substituted  in  y=f(x),  gives  imaginary,  or 
negative  values  of  y,  the  result  also  may  have  no  engineering 
application.  In  still  other  cases,  the  mathematical  result 
may  give  values,  which  are  so  far  beyond  the  range  of  indus- 
trially practicable  numerical  values  as  to  be  inapplicable. 
For  instance : 

Example  9.  In  example  (8),  to  determine  the  resistance 
r,  which  gives  maximum  current  transmitted  over  a  trans- 
mission line,  the  (Miuation  (15), 


1  = 


\  (/■  I  '■„)-  I  .r,r' 


MAXIMA   AND  Ml  MM  A.  161 

immediately  differentiated,  gives  as  condition  of  the  extremes: 

(M= 2(r+r0) 

dr         ■2\(r  +  ro)2+xo2\V(r  +  r0y2+x?~    ' 

hence,  either  r  +  ro  =  0; (22) 

or,  (r  +  r0)2+xo2  =  oc (23) 

the  latter  equation  gives  r  =  ao;  hence  i  =  Q,  the  minimum  value 
of  current. 

The  former  equation  gives 

r=-r0, (24) 

as  tne  value  of  the  resistance,  which  gives  maximum  current, 
and  the  current  would  then  be,  by  substituting  (24)  into  (15), 

i=j (25) 

Xq 

The  solution  (24),  however,  has  no  engineering  meaning, 
as  the  resistance  r  cannot  be  negative. 

Hence,  mathemetically,  there  exists  no  maximum  value 
of  i  in  the  range  of  r  which  can  occur  in  engineering,  that  is, 
within  the  range,  0<  r<  -x>. 

In  such  a  case,  where  the  extreme  falls  outside  of  the  range 
of  numerical  values,  to  which  the  engineering  quantity  is 
limited,  it  follows  that  within  the  engineering  range  the  quan- 
tity continuously  increases  toward  one  limit  and  continuously 
decreases  toward  the  other  limit,  and  that  therefore  the  two 
limits  of  the  engineering  range  of  the  quantity  give  extremes. 
Thus  r=  Ogives  the  maximum,  r  =  oothe  minimum  of  current. 

106.  Example  io.  An  alternating-current  generator,  of 
generated  e.m.f.  e  =  2500  volts,  internal  resistance  r0  =  0.25 
ohms,  and  synchronous  reactance  x0  =  10  ohms,  is  loaded  by 
a  circuit  comprising  a  resistor  of  constant  resistance  r  =  20 
ohms,  and  a  reactor  of  reactance  x  in  series  with  the  resistor 
r.     What  value  of  reactance  x  gives  maximum  output? 

If  i  =  current  of  the  alternator,  its  power  output  is 

P  =  n'2  =  20i2; (20) 


1(12  ENGINEERING    MATHEMATICS. 

the  total  resistance  is  M  r0    20.25  ohms;    the  total  reactance 
is  .r+x0  =  10+.r  ohms,  and  therefore  the  current  is 


V(r+r0)2  +  (x  +  x0)2 


(27) 


and  the  power  output,  by  substituting  (27)  in  (20),  is 

re*  20X2500* 

~(r+r02)+(x+xo)2     20.252  +  (10+z)2'  *     *     v" 

Simplified,  this  gives 

yi  =  (r+ro)2  +  (x+x0)2; (29) 

2/2'-=(-r+-*"o)2; 


hence, 


:2(.r+.r0)=0; 


dx 
and 

x=-.r0=-10ohms; (30) 

that  is,  a  negative,  or  condensive  reactance  of  10  ohms.     The 
power  output  would  then  be,  by  substituting  (30)  into  (28), 

_         re2         20+25002       ++       _n_  f     . 

P==¥^orT=    20.252     watts=305kw-    •    •     (31) 

If,  however,  a  condensive  reactance  is  excluded,  that  is, 
it  is  assumed  that  x>0,  no  mathematical  extreme  exists  in  the 
range  of  the  variable  x,  which  is  permissible,  and  the  extreme 
is  at  the  end  of  the  range,  x  =  0,  and  gives 

P=7-T^£l—  =  245  kw (32) 

107.  Example  11.  In  a  500-kw.  alternator,  at  voltage 
e  =  2500,  the  friction  and  windage  loss  is  Pw  =  6  kw.,  the  iron 
loss  Pt  =  24  kw.,  the  field  excitation  loss  is  Pf  =  (\  kw.,  and  the 
armature  resistance  r  =  0.1  ohm.  At  what  load  is  the  efficiency 
a  maximum? 


MAXIMA    AND  MINIMA.  163 

The  sum  of  the  losses  is: 

P  =  Pw  +  P{  +  Pf  +  r/2-  3(3,000  +0.1?2.     .     .     .     (33) 

The  output  is 

P0  =  d=2500i; (34) 

hence,  the  efficiency  is 

Po     = ei  = 2500^ 

'     Po  +  P    ei+Pw+Pi+Pf+ri2    36000+25001+0.11*'     (     } 


or,  simplified, 


hence, 


Pw+Pi+P/i     . 

2/i=-      —> +n; 


^i         Pw  +  Pj  +  Pf 
di     r~  i2 


and, 


^  =  a|-  "a/-7TT     =600  amperes,  (36) 


/360D0 

0.1 


and  the  output,  at  which  the  maximum  efficiency  occurs,  by 
substituting  (36)  into  (34),  is 

P  =  ei=1500  kw., 

that  is,  at  three  times  full  load. 

Therefore,  this  value  is  of  no  engineering  importance,  but 
means  that  at  full  load  and  at  all  practical  overloads  the 
maximum  efficiency  is  not  yet  reached,  but  the  efficiency  is 
still  rising. 

108.  Frequently  in  engineering  calculations  extremes  of 
engineering  quantities  are  to  be  determined,  which  are  func- 
tions or  two  or  more  independent  variables.  For  instance, 
the  maximum  power  is  required  which  can  be  delivered  over  a 
transmission  line  into  a  circuit,  in  which  the  resistance  as  well 
as  the  reactance  can  be  varied  independently.  In  other 
words,  if 

y=f(u,v) (37) 


164  ENGINEERING  MA  THEM  AT H  'S. 

is  a  function  of  two  independent  variables  u  and  v,  such  a 
pair  of  values  of  u  and  of  v  is  to  be  found,  which  makes  y  a 
maximum,  or  minimum. 

Choosing  any  value  u0,  of  the  independent  variable  u, 
then  a  value  of  v  can  be  found,  which  gives  the  maximum  (or 
minimum)  value  of  y,  which  can  be  reached  for  u  =  u0.  This 
is  done  by  differentiating  y=f(u0,v),  over  v,  thus: 

dfiy]  =0, (38) 

dv 

From  this  equation  (38),  a  value, 

v=Mu0), (39) 

is  derived,  which  gives  the  maximum  value  of  y,  for  the  given 
value  of  wo,  and  by  substituting  (39)  into  (38), 

y=f2(uo), (40) 

is  obtained  as  the  equation,  which  relates  the  different  extremes 
of  ?/,  that  correspond  to   the  different  values  of  u0,  with    u0. 
Herefrom,  then,  that  value  of  u0  is  found  which  gives  the 
maximum  of  the  maxima,  by  differentiation: 

■      ■        *^L0 m 

duo 

Geometrically,  y=f(u,r)  may  be  represented  by  a  surface 
in  space,  with  the  coordinates  y,  u,  v.  y  =f(uo,v),  then,  represents 
the  curve  of  intersection  of  this  surface  with  the  plane  uq  = 
constant,  and  the  differentation  gives  the  maximum  point 
of  this  intersection  curve.  2/=/2(^o)  then  gives  the  curve 
in  space,  which  connects  all  the  maxima  of  the  various  inter- 
sections with  the  uq  planes,  and  the  second  differentiation 
gives  the  maximum  of  this  maximum  curve  ;/=f-2(uo),  or  the 
maximum  of  the  maxima  (or  more  correctly,  the  extreme  of 
the  extremes). 

Inversely,  it  is  possible  first  to  differentiate  over  u,  thus, 

df(u,v0)_ 


MAXIMA    AND  MINIMA.  105 

and  thereby  get 

u=/s(»o), (43) 

as  the  value  of  u,  which  makes  y  a  maximum  for  the  given 
value  of  v  =  Vo,  and  substituting  (43)  into  (42), 

y=Mv0), (44) 

is  obtained  as  the  equation  of  the  maxima,  which  differentiated 

over  Vo,  thus, 

<lrn 


=  0,    .     . (45) 


gives  the  maximum  of  the  maxima. 

Geometrically,  this  represents  the  consideration  of  the 
intersection  curves  of  the  surface  with  the  planes  v  =  constant. 

However,  equations  (38)  and  (41)  (respectively  (42)  and 
(45))  give  an  extremum  only,  if  both  equations  represent 
maxima,  or  both  minima.  If  one  of  the  equations  represents 
a  maximum,  the  other  a  minimum,  the  point  is  not  an  extre- 
mum, but  a  saddle  point,  so  called  from  the  shape  of  the  sur- 
face y=f(u,  v)  near  this  point. 

The  working  of  this  will  be  plain  from  the  following  example: 

109.  Example  12.  The  alternating  voltage  e  =  30,000  is 
impressed  upon  a  transmission  line  of  resistance  ro  =  20  ohms 
and  reactance  .r0  =  50  ohms. 

What  should  be  the  resistance  r  and  the  reactance  x  of  the 
receiving  circuit  to  deliver  maximum  power? 

Let  i  =  current  delivered  into  the  receiving  circuit.  The 
total  resistance  is  (r  +  r0);  the  total  reactance  is  (x+Xo);  hence, 
the  current  is 

....     (46) 


V(r+7-0)2  +  (.r+.r0)2 

The  power  output  is         P=ri2;        (47) 

hence,  substituting  (40),  gives 

r&  

(r  +  r0)2  +  (x+x0)2 

(a)  For  any  given  value  of  r,  the  reactance  x,  which  gives 

dP 
maximum  power,  is  derived  by  t~=0. 


166  ENGINEERING  MATHEMATICS.         , 

P  simplified,  gives  y\  =  (x  +x0)2;  hence, 

dii\ 
*-=2(x+x0)=Q    and     x=-x0    .     .     .     (49) 


dx 


that  is,  for  any  chosen  resistance  r,  the  power  is  a  maximum, 
if  the  reactance  of  the  receiving  circuit  is  chosen  equal  to  that 
of  the  line,  but  of  opposite  sign,  that  is,  as  condensive  reactance. 
Substituting  (49)  into  (48)  gives  the  maximum  power 
available  for  a  chosen  value  of  r,  as : 


re2 


or,  simplified, 


Po  =  R7^       <*» 


M      (r  +  r0)2  r02 

2/2  =  — — -    and      7/3  =  r+—; 


hence, 

~cfr=1~~r2       and        r  =  rQ;     ....     (51) 
and  by  substituting  (51)  into  (50),  the  maximum  power  is, 

e2 

max  ~  Aj,   • l^-) 

(b)  For  any  given  value  of  x,  the  resistance  r,  which  gives 

dP 
dr 


dP 
maximum  power,  is  given  by  -=—  =0. 


P  simplified  gives, 

w        (?-+r0)2  +  (x+x0)2  r2  +  (x+x)2 

r  r 

dy2     1  jro2  +  (x+.r0)2 
dr         +  r2         ^~U- 


r=\/r02  +  (x+x0)2,        (53) 

which  is  the  value  of  r,  that  for  any  given  value  of  x,  gives 
maximum  power,  and  this  maximum  power  by  substituting 
(53)  into  (48)  is, 

p  \V  +  (x  +  .r„)-V- 

*  o  =  ~ 


I/o  +  Vr02  +  (x+^f+  {x+xo)2 

e2 

2fr0  +  v/^  +  (x+a;o)2|; (54) 


MAXIMA   AND  MINIMA.  107 

which  is  the  maximum  power  that  can  be  transmitted  into  a 
receiving  circuit  of  reactance  x. 

The  value  of  x,  which  makes  this  maximum  power  Pothe 

highest  maximum,  is  given  by  -r—=Q. 
Po  simplified  gives 


ys  =  r0  +  \/ro2  +  (x+xo)2; 


ij4=Vro-  +  {x+x0y2; 
y5  =  ro2  +  (x+x0)2; 
y(i  =  (x+xo)2; 
y7  =  (x+x0); 
and  this  value  is  a  maximum  for  (.r  +  .ro)=0;    that  is,  for 

x  =  -  xq (55) 

Note.  If  x  cannot  be  negative,  that  is,  if  only  inductive 
reactance  is  considered,  x  =  0  gives  the  maximum  power,  and 
the  latter  then  is 


max 


2|r0  +  Vro2  +  .r02V 


•     (56) 


the  same  value  as  found  in  problem  (7),  equation  (18). 

Substituting  (55)  and  (54)  gives  again  equation  (52),  thus, 


p   -*- 

x  max        \  „.   • 
"i'O 

no.  Here  again,  it  requires  consideration,  whether  the 
solution  is  practicable  within  the  limitation  of  engineering 
constants. 

With  the  numerical   constants  chosen,  it  would  be 

e2      300002 
^>max  =  ^r=—g0-  -=H,250  kw.; 


e 
■■r—  =  750  amperes, 


L68  ENGINEERING  MATHEMA  TICS. 

and  the  voltage  at  the  receiving  end  of  the  line  would  be 


et  =  ix/f>  +  X2  =  750  v  2U2  +  502  =  40,400  volts ; 

that  is,  the  voltage  at  the  receiving  end  would  be  far  higher 
than  at  the  generator  end,  the  current  excessive,  and  the  efficiency 
of  transmission  only  50  per  cent.  This  extreme  case  thus  is 
hardly  practicable,  and  the  conclusion  would  be  that  by  tin; 
use  of  negative  reactance  in  the  receiving  circuit,  an  amount 
of  power  could  be  delivered,  at  a  sacrifice  of  efficiency,  far 
greater  than  economical  transmission  would  permit. 

In  (lie  case,  where  capacity  was  excluded  from  the  receiv- 
ing circuit,  the  maximum  power  was  given  by  equation  (50)  as 

7  max  =-7-—4==^  =  0100  kw. 

2jr0  +  \/r0-  +  .ro-! 

in.  Extremes  of  engineering     quantities  x,  y,  are  usually 
determined  by  differentiating  the  function, 


and  from  the  equation, 


y=f(x), (57) 


2-o w 


deriving  the  values  oi  x,  which  make  y  an  extreme. 

Occasionally,  however,  the  equation  (58)  cannot  be  solved 
for  x,  but  is  either  of  higher  order  in  x,  or  a  transcendental 
equation.  In  this  case,  equation  (58)  may  be  solved  by  approx- 
imation, or  preferably,  the  function, 

z=^ (59) 

ax, 

is  plotted  as  a  curve,  the  values  of  x  taken,  at  which  2  =  0, 

that  i-,  :it  which  the  curve  intersects  the  X-axis.     For  instance: 

Example    13.    Tin;    e.ni.f.  wave    of  a  three-phase  alternator, 

as  determined  by  oscillograph,  is  represented  by  the  equation, 

e=36000{s'm  #-0.l2sin  (30  -  2.3°)-23sin  (50-1.5°)  + 

0.13  sin  (70-6.2°)} (00) 


MAXIMA  AND  MIX  IMA 


169 


This  alternator,  connected  to  a  long-distance  transmission  line, 
gives  the  charging  current  to  the  line  of 

i  =  13.12  cos  (0-O.3°)-5.O4  cos  (30-3.3°) -18.76  cos  (50-3.6°) 

+  19.59  cos  (70-9.9°)     ....     (01) 

(see  Chapter  III,  paragraph  95). 

What  arc  the  extreme  values  of  this  current,  and  at  what 
phase  angles  6  do  they  occur? 

The  phase  angle  0,  at  which  the  current  i  reaches  an  extreme 
value,  is  given  by  the  equation 


di 
dd 


=  0. 


(62) 


Fig.  57 


Substituting  (61)  into  (62)  gives, 

z=Jq=  - 13.12  sin  (0-0.3°)  +15.12  sin  (30-3.3°)  +93.8  sin 

(50- 3.6°) -137.1  sin  (70-9.9°)  =0.     .     .     .     (63) 

This  equation  cannot  be  solved  for  0.  Therefore  z  is 
plotted  as  function  of  0  by  the  curve,  Fig.  57,  and  from  this 
curve  the  values  of  0  taken  at  which  the  curve  intersects  the 
zero  line.     They  are: 

0  =  1°;  20°;  47°  78°;  104°;  135°;  162°. 


170  i-:s(;L\KEinsa  ma t hematics. 

For  these  angles  6,  the  corresponding  values  of  i  are  calculated 
by  equation  (01),  and  arc: 

?o=  +9;    -1;   +39;    -30;    +30;    -42;    +4 amperes. 

The  current  thus  has  during  each  period  14  extrema,  of 
which  the  highest  is  42  amperes. 

ii2.  In   those   cases,   where   the   mathematical   expression 

of  the  function  y=f(x)  is  not  known,  and  the  extreme  values 

therefore  have  to  be  determined  graphically,  frequently  a  greater 

accuracy  can  be  reached  by  plotting  as  a  curve  the  differential 

of  y=f(x)  and  picking  out  the  zero  values  instead  of  plotting 

y=f(x),  and  picking  out  the  highest  and  the  lowest   points. 

If  the  mathematical  expression  of  y=f(x)  is  not  known,  obvi- 

dii 
ously  the  equation  of  the  differential  curve  z=—  (04)  is  usually 

ax 

not  known  either.  Approximately,  however,  it  can  fre- 
quently be  plotted  from  the  numerical  values  of  y=f(x),  as 
follows : 

If      X\,  x->,  xs  .  .  .  are  successive  numerical  values  of  x, 

and         2/i,  y-j,  Vz  ■  ■  ■  the  corresponding  numerical  values  of  y, 

approximate  points  of  the  differential  curve  z=~r  are  given 
by  the  corresponding  values: 

X2+X}  XS+X2  X4+X3 


as  abscissas: 


as  ordinates: 


2      '  2      '  2 

V2-yi ,    j/3-2/2 .    va-v-a 

X2~X\  '      Xz  —  X2  '       £4—  £3' 


113.  Example  14,  In  the  problem  (1),  the  maximum  permea- 
bility point  of  a  sample  of  iron,  of  which  the  (B,  3C  curve  is  given 
as  Fig.  51,  was  determined  by  taking  from  Fig.  51  corresponding 

values  of  CB  and  3C,  and  plotting  //=— ,  against  CB  in    Fig.   52. 

A  considerable  inaccuracy  exists  in  this  method,  in  locating 
,ni  value  of  CB,  ;ii  which  fx  is  a  maximum,  due  to  the  flatness 
of  the  curve,  Fig.  52. 


MAXIMA   AND  MINIMA. 


171 


The  successive  pairs  of  corresponding  values  of  G$  and  3C, 

as  taken  from  Fig.  51  arc  given  in  columns  1  and  2  of  Table  I. 

Table   I. 


05 

Kilolines, 

3C 

(B 

Ay. 

03 

0 

1 

2 

0 

1.76 
2.74 

370 
570 
730 

+  200 
160 

0.5 
1.5 

3 
4 
5 

3.47 
4.06 

4 .  59 

•  S65 

MS.-, 

1090 

135 
120 
105 

2.5 
3.5 
4.5 

6 

7 
8 

5.10 
5.63 
6.17 

1175 
1245 
1295 

85 
70 
50 

5.5 
6.5 
7.5 

9 
10 
11 

6.77 
7.47 
8.33 

1330 
1340 
1320 

35 
+  10 
-  20 

8.5 

9.5 

10.5 

12 
13 
14 

9.60 
11.60 
15.10 

1250 

1120 

930 

70 
130 
190 

11.5 
12.5 
13.5 

15 

20.7 

725 

205 

14.5 

In  the  third  column  of  Table  I  is  given  the  permeability, 

/r> 

[i  =— ,  and  in  the  fourth  column  the  increase  of  permeabilitv, 

per  (B=l,  J/t;  the  last  column  then  gives  the  value  of  03,  to 
which  J/j.  corresponds. 

In  Fig.  58,  values  A/i  are  plotted  as  ordinates,  with  03  as 
abscissas.     This  curve  passes  through  zero  at  03  =  9.95. 

The  maximum  permeability  thus  occurs  at  the  approximate 
magnetic  density  03  =  9.95  kilolines  per  sq.cm.,  and  not  at  03  = 
10.2,  as  was  given  by  the  less  accurate  graphical  determination 
of  Fig.  52,  and  the  maximum  permeability  is  /to  =  1340. 

As  seen,  the  sharpness  of  the  intersection  of  the  differential 
curve  with  the  zero  line  permits  a  far  greater  accuracy  than 
feasible  by  the  method  used  in  instance  (1). 

ii4-  As  illustration  of  the  method  of  determining  extremes, 
some  further  examples  are  given  below : 


172 


ENGINEERING  MA  THEM  A  TICS. 


Example  15.  A  .storage  battery  of  n  =  80  cells  is  to  be 
connected  so  as  to  give  maximum  power  in  a  constant  resist- 
ance r  =  0.1  ohm.  Each  battery  cell  has  the  e.m.f.  e()  =  2.1 
volts  and  the  internal  resistance  ro  =  0.02  ohm.  How  must 
the  cells  be  connected? 

Assuming  the  cells  are  connected  with  x  in  parallel,  hence 

71 

—  in  series.     The  internal  resistance   of  the   battery  then  is 

JO 

n 

=— 7-  ohms,  and  the  total  resistance  of  the  circuit  is  —,r0  +  r. 

x        x2  x2 


Fig.  5S.     First  Differential  Quotient  of  (B,/<  Curve 

Tl  71 

The  e.m.f.  acting  on  the  circuit  is  —  eo,  since  -  cells  of  e.m.f. 

eo  arc  in  scries.     Therefore,  the  current  delivered  by  the  battery 
is, 

n 


n 

x2 


r0+r 


and  the  power  which  this  current  produces  in  the  resistance 
r,  is, 


P=ri2  = 


rn2('tr 


x*  { '',  /■„  I  /•) 


MAXIMA  AND  MINIMA.  173 

This  is  an  extreme,  if 


is  an  extreme,  hence, 


and 


nr0  , 
w= Yrx 

u      x 


ax         xz 


that    is,    .r  =  ^/~-7!-  =  4    cells    are    connected    in    multiple,    and 

71  If  IT 

— =*/ —  =  20  cells  in  series. 
x      \  r0 

115.  Example  16,  In  an  alternating-current  transformer  the 
loss  of  power  is  limited  to  900  watts  by  the  permissible  temper- 
ature rise.  The  internal  resistance  of  the  transformer  winding 
(primary,  plus  secondary  reduced  to  the  primary)  is  2  ohms, 
and  the  core  loss  at  2000  volts  impressed,  is  400  watts,  and 
varies  with  the  1.6th  power  of  the  magnetic  density  and  there- 
fore of  the  voltage.  At  what  impressed  voltage  is  the  output 
of  the  transformer  a  maximum? 

If  e  is  tht;  impressed  e.m.f.  and  i  is  the  current  input,  the 
power  input  into  the  transformer  (approximately,  at  non- 
inductive  load)  is  P  =  ei. 

If  the  output  is  a  maximum,  at  constant  loss,  the  input  P 
also  is  a  maximum.  The  loss  of  power  in  the  winding  is 
r/-  =  2i2. 

The  loss  of  power  in  the  iron  at  2000  volts  impressed  is 
100  watts,  and  at  impressed  voltage  e  it  therefore  is 

(mo)  "°x400' 

and  the  total  loss  in  the  transformer,  therefore,  is 
Pi-2t*+400yj5V'6  =  900; 


1 7  1  ENGINEERING   M  - 1  THEM.  \  TI(  '8. 

herefrom,  it  follows  that, 


i-^so-aoo^)'", 

and,  substituting,  into  P=ei: 


/      e     \  1-6 


P=eM50-200{mQ 


e3-6 


Simplified,  this  gives, 

y  — •*     20001-5' 

and,  differentiating, 

dy_  3.6g2'6 

de      '5e    20001<s    U> 

and 

\2000/ 

Hence, 

-^  =  1.15    and     e= 2300  volts, 
which,  substituted,  gives 


P  =  2300  \/450-200X  1  .25  =  32.52  kw. 

116.  Example  17.  In  a  5-kw.  alternating-current  transformer, 
at  1000  volts  impressed,  the  core  loss  is  GO  watts,  the  i2r  loss 
150  watts.  How  must  the  impressed  voltage  be  changed, 
to  give  maximum  efficiency,  (a)  At  full  load  of  5-kw;  (6)  at 
half  load? 

The  core  loss  may  be  assumed  as  varying  with  the  1.6th 
power  of  the  impressed  voltage.     If  e  is  the  impressed  voltage, 

1  =  is  the  current  at  lull  load,  and ii  = is  the  current  at 

e  e 

half  load,  then  at  1000  volts  impressed,  the  full-load  current  is 
-  =  5  amperes,  and  since  the  i-r  loss  is  150  watts,  this  gives 


MAXIMA  AND  MINIMA.  175 

the  internal  resistance  of  the  transformer  as  r  =  6  ohms,  and 
herefrom  the  i2r  loss  at  impressed  voltage  e  is  respectively, 

•     150  XlO6  ,       ._    37.5  XlO6 

n2  = ^ and    nr  = s wat  t  s . 

e2  e2 

Since  the  core  loss  is  60  watts  at  1000  volts,  at  the  voltage  e 
it  is 

P;=60x(j~)      watts. 

The  total  loss,  at  full  load,  thus  is 

„       n        .n  /    e    y-G     150X10° 

and  at  half  load  it  is 

P       P    ,n-o     gyj    '    V°|37.5X10« 
Pn-fi+ni  -60xVl000y  e2       ' 

Simplified,  this  gives 

/  e  y-6 

2/l-(I^5)1'6+O.625Xl0exe-; 

hence,  differentiated, 

L6l^-6-5xl06e_3=05 

pO-0 

Lr)Tooo^"L25xlOGx^3=0; 

e3-6  =  3.125  X106X10001'6  =  3.125  XlO10-8; 

e3-6  =  0.78125  X 106  X 10001-0  =  0.78125  XlO10-8 ; 

hence,        e  =  1373  volts  for  maximum  efficiency  at  full  load. 

and  e  =  938  volts  for  maximum  efficiency  at  half  load. 

117.  Example  18.  (a)  Constant  voltage  e  =  1000  is  im- 
pressed upon  a  condenser  of  capacity  C  =  10  mf.,  through  a 
reactor  of  inductance  L  =  100  mh.,  and  a  resistor  of  resist- 
ance  r  =  40  ohms.  What  is  the  maximum  value  of  the  charg- 
ing current? 


176  ENGINEERING   MATHEMATICS. 

{In  An  additional  resistor  of  resistance  r'  =  210  ohms  is 
then  inserted  in  series,  making  the  total  resistance  of  the  con- 
denser charging  circuit,  r  =  250  ohms.  What  is  the  maximum 
value  of  the  charging  current? 

The  equation  of  the  charging  current  of  a  condenser,  through 
a  circuit  of  low  resistance,  is  ("Transient  Electric  Phenomena 
and  Oscillations,"  p.  61): 


where 


i  =  ~\e   aL'sinXJ, 
q  [  2L\' 


\\L 


c    7'2' 


and  the  equation  of  the  charging  current  of  a  condenser,  through 
a  circuit  of  high  resistance,  is  ("  Transient  Electric  Phenomena 
and  Oscillations,"  p.  51), 

s  [  \> 

where 


4 


4L 

— 77- 


Substituting  the  numerical  values  gives: 

(a)  i=10.2  £-200isin  980  /; 

(6)  i=6.667j  e~500t-  ff-20oo«| 

Simplified  and  differentiated,  this  gives: 

W  z=~dt  =   l-9  c°s  980«-sin  980*=0; 

hence  tan980f=4.9 

980*  =  68.5°       =1.20 
.     1.20  +nn 


DOS 


m 


(b)  Z=  =4£-2000/_  c-500/=A. 

///  w  v'  • 


ill 


MAXIMA   AND  MINIMA.  177 


hence,  £+i5oo,  =  4j 

log  4 
1500*  =  ,——  =  1.38, 

log:  s 

t  =  0.00092  sec, 

and,  by  substituting  these  values  of  /  into  the  equations  of  the 
current,  gives  the  maximum  values: 

1.20+T17T 

(a)       i=10e       4.9     =7.S3£-°-64w  =  7.83X0.53"  amperes; 

that  is,  an  infinite  number  of  maxima,  of  gradually  decreasing 
values:  +7.83;    -4.1.5;    +.2.20;    -1.17  etc. 

(&)  i  =().6(>7(e-°-46-  £~1-84)=3.16  amperes. 

118.  Example  19.  In  an  induction  generator,  the  fric- 
tion losses  are  Py=100  kw.;  the  iron  loss  is  200  kw.  at  the  ter- 
minal voltage  of  e  =  4  kv.,  and  may  be  assumed  as  proportional 
to  the  1.6th  power  of  the  voltage;  the  loss  in  the  resistance 
of  the  conductors  is  100  kw.  at  i  =  3000  amperes  output,  and  may 
be  assumed  as  proportional  to  the  square  of  the  current,  and 
the  losses  resulting  from  stray  fields  due  to  magnetic  saturation 
are  100  kw.  at  e  =  4  kv.,  and  may  in  the  range  considered  be 
assumed  as  approximately  proportional  to  the  3.2th  power 
of  the  voltage.  Under  what  conditions  of  operation,  regard- 
ing output,  voltage  and  current,  is  the  efficiency  a  maximum? 

The  losses  may  be  summarized  as  follows: 

Friction  loss,  P/=100  kw.; 

Iron  loss,  P,+20o(| 


^rloss,  Pc  =  1oo(^)2; 

/e\3.2 

Saturation  loss,  Ps  =  100(j)      ; 

hence  the  total  loss  is  PL  = P/+  P;  +PC  +  PS 


ITS  ENGINEERING   MATHEMATICS. 

The  output  is  P=ei;  hence,  percentage  of  loss  is 

e\1.6       /      i     \2        /e\3.2] 


The  efficiency  is  a  maximum,  if  the  percentage  loss  X  is  a 
minimum.      For  any  value  of  the  voltage  e,  this  is  the  case 

at  the  current  i,  given  by  "7^=0;  hence,  simplifying  and  differ- 
entiating I, 


dl         1+2W      +  V4  7  1 


'onnn2      vf 


di  i2  3000 

i=3000jl+2(|)1"6  +  (|)3'2; 

then,  substituting  i  in  the  expression  of  X,  gives 

and  A  is  an  extreme,  if  the  simplified  expression, 

2/=^2   '  41-6^0-4  "■  43-2e 

is  an  extreme,  at 

<fy_  _2         0.8       O 
de~      e3    41-6e1'4+43'2 

0.8  1.: 

41-6°        43 


hence,  2+7^6^- ^e»-»-0; 


/g\Mi         2 

hence,  (-)      =—     and     e  =  5.50  kv., 

and,  by  substitution  the  following  values  are  obtained  :X= 0.0323; 
efficiency  96.77  per  cent;  current  i=8000  amperes:  output 
I'      1 1,000  k\v. 

119.  In  all  probability,  this  output  is  beyond  the  capacity 

of  the  generator,  as  limited  by  heating.  The  foremost  limita- 
tion probably  will  be  the  i2r  heating  of  the  conductors;   that  is, 


MAXIMA  AND  MINIMA.  179 

the   maximum   permissible   current  will   be   restricted  to,   for 

instance,  i  =  5000  amperes. 

For  any  given  value  of  current  i,  the  maximum  efficiency, 
that  is,  minimum  loss,  is  found  by  differentiating:, 

A  —  . 

ei 
over  e,  thus : 

dk  _ 
de 

Simplified,  /  gives 

1  f        /    i    \21        2  I 

hence,  differentiated,  it  gives 

dy_      If       (_J_\2\        1-2       2.2c1-2 
de~     e2\       \3000/  J  +41"6e°-4+  4?-2         ' 
e\3.2     6/ey-8     5  f  .      /    ''    \21 

1/  +n(v4J  =nt1+lv3ooo;  |; 

e\i-«     ~3+VG1+'55  3000 


U/  11 

For  1  =  5000,  this  gives: 


e  \  i-6 


=  1.065    and    c  =  4.16  kv.; 

hence, 

A  =  0.0338,  Efficency  96.62  per  cent,  Power  P=  20,800  kw. 

Method  of  Least  Squares. 

120.  An  interesting  and  very  important  application  of  the 
theory  of  extremes  is  given  by  the  method  of  least  squares,  which 
is  used  to  calculate  the  most  accurate  values  of  the  constants 
of  functions  from  numerical  observations  which  are  more  numei- 
ous  than  the  constants. 

if  y=M, (i) 


1 81  >  ENGINEERING  MATHEMATICS. 

is  a  function  having  the  constants  a,  b,  c  .  .  .  and  the  form  of 
the  function  (1)  is  known,  for  instance, 

y  =  a  +  bx+cx2, (2) 

and  the  constants  a  b,  c  are  not  known,  but  the  numerical 
values  of  a  number  of  corresponding  values  of  x  and  y  are  given, 
for  instance,  by  experiment,  X\,  x2,  x3,  x4  .  .  .  and  y\,  y2, 2/3, 2/4  •  •  •  , 
then  from  these  corresponding  numerical  values  x„  and  yn 
the  constants  a,  b,  c  .  .  .  can  be  calculated,  if  the  numerical 
values,  that  is,  the  observed  points  of  the  curve,  arc  sufficiently 
numerous. 

If  less  points  x\  y\,  x2,  y2  ■  ■  •  are  observed,  then  the  equa- 
tion (1)  has  constants,  obviously  these  constants  cannot  be 
calculated,  as  not  sufficient  data  are  available  therefor. 

If  the  number  of  observed  points  equals  the  number  of  con- 
stants, they  an;  just  sufficient  to  calculate  the  constants.  For 
instance,  in  equation  (2),  if  three  corresponding  values  x\,  y\) 
•'_-.  y-2',  -i';i,  Vz  are  observed,  by  substituting  these  into  equation 
(2),  three  equations  are  obtained: 

?/i  =  a  +  6xi  +cxi2; 

y2  =  a-\-bx2+cx22;  j- (3) 

y3  =  a:i+bx+cxs2, 

which  are  just  sufficient  for  the  calculation  of  the  three  constants 
a,  b,  c. 

Three  observations  would  therefore  be  sufficient  for  deter- 
mining three  constants,  if  the  observations  were  absolutely 
correct.  This,  however,  is  not  the  case,  but  the  observations 
always  contain  errors  of  observation,  that  is,  unavoidable  inac- 
curacies, and  constants  calculated  by  using  only  as  many 
observations  as  there  are  constants,  are  not  very  accurate. 

Thus,  in  experimental  work,  always  more  observations 
are  made  than  just  necessary  for  the  determination  of  the 
constants,  for  the  purpose  of  getting  a  higher  accuracy.  Thus, 
for  instance,  in  astronomy,  for  the  calculation  of  the  orbit  of 
:i  comet,  less  than  four  observations  are  theoretically  sufficient, 
but  if  possible  hundreds  are  taken,  to  get  a  greater  accuracy 
in  the  determination  of  the  constants  of  the  orbit. 


MAXIMA  AND  MINIMA.  181 

If,  then,  for  the  determination  of  the  constants  a,  b,  c  of 
(Miuation  (2),  six  pairs  of  corresponding  values  of  x  and  y  were 
determined,  any  three  of  these  pairs  would  be  sufficient  to 
give  a,  b,  c,  as  seen  above,  but  using  different  sets  of  three 
observations,  would  not  give  the  same  values  of  a,  b,  c  (as  it 
should,  if  the  observations  were  absolutely  accurate),  but 
different  values,  and  none  of  these  values  would  have  as  high 
an  accuracy  as  can  be  reached  from  the  experimental  data, 
since  none  of  the  values  uses  all  observations. 

I21-  ^  y=JV), (i) 

is  a  function  containing  the  constants  a,  b,  c  .  .  .,  which  are  still 
unknown,  and  xi,  y\)  x2,  y2;  x3,  y3;  etc.,  are  corresponding 
experimental  values,  then,  if  these  values  were  absolutely  cor- 
rect, and  the  correct  values  of  the  constants  a,b,  c  .  .  .  chosen, 
?/i=/(^i)  would  be  true;    that  is, 


f(x1)-y1=0; 
f(x2)  —  y 2  =  0,  etc. 


(5) 


Due  to  the  errors  of  observation,  this  is  not  the  case,  but 
even  if  a,  b,  c  .  .  .  are  the  correct  values, 

?/i  ^/(xi)  etc.; (6) 

that  is,  a  small  difference,  or  error,  exists,  thus 

fM-yi  =  di)  ] 

\ (7) 

f(x2)- y2  =  d2,  etc.;  j 


If  instead  of  the  correct  values  of  the  constants,  a,  b,  c  .  .  ., 
other  values  were  chosen,  different  errors  u\,  d2  .  .  .  would 
obviously  result. 

From  probability  calculation  it  follows,  that,  if  the  correct 
values  of  the  constants  a,  b,  c  .  .  .  are  chosen,  the  sum  of  the 
squares  of  the  errors, 

^i2  +  oV  +  oV  + (8) 

is  less  than  for  any  other  value  of  the  constants  a,  b,  c  .  .  .;  that 
is,  it  is  a  minimum. 


1 82  ENGINEERING  MA  THEMATICS. 

122.  The  problem  of  determining" the  constants  a,  b,  c.  .., 
thus  consists  in  finding  a  set  of  constants,  which  makes  the 
sum  of  the  squares  of  the  errors  3  a  minimum;    that  is, 

z=  Hd2= minimum, (9) 

is  the  requirement,  which  gives  the  most  accurate  or  most 
probable  set  of  values  of  the  constants  a,  b,  c  .  .  . 

Since  by  (7),  d=f(x)  —  y,  it  follows  from  (9)  as  the  condi- 
tion, which  gives  the  most  probable  value  of  the  constants 
a,  b,  c  .  .  . ; 

z=H\f(x)  — 1/}2  =  minimum;      ....     (10) 

that  is,  the  least  sum  of  the  squares  of  the  errors  gives  the  most 
probable  value  of  the  constants  a,  b,  c .  .  . 

To  find  the  values  of  a,  b,  c  .  .  .,  which  fulfill  equation  (10), 
the  differential  quotients  of  (10)  are  equated  to  zero,  and  give 

dz     sr^    ,.,  s        ,df(x) 


^-2.^)-^}-^— 0;etc. 


(11) 


This  gives  as  many  equations  as  there  are  constants  a,  b,  c  . . ., 
and  therefore  just  suffices  for  their  calculation,  and  the  values 
so  calculated  are  the  most  probable,  that  is,  the  most  accurate 
values. 

Where  extremely  high  accuracy  is  required,  as  for  instance 
in  astronomy  when  calculating  from  observations  extending 
over  a  few  months  only,  the  orbit  of  a  comet  which  possibly 
lasts  thousands  of  years,  the  method  of  least  squares  must  be 
used,  and  is  frequently  necessary  also  in  engineering,  to  get 
from  a  limited  number  of  observations  the  highest  accuracy 
of  the  constants. 

123.  As  instance,  the  method  of  least  squares  may  be  applied 
in  separating  from  the  observations  of  an  induction  motor, 
when  running  light ,  the  component  losses,  as  friction,  hysteresis, 
etc. 


MAXIMA   AND  MINIMA. 


183 


In  a  440-volt  50-h.p.  induction  motor,  when  running  light, 
that  is,  without  load,  at  various  voltages,  let  the  terminal 
voltage  e,  the  current  input  i,  and  the  power  input  p  be  observed 
as  given  in  the  first  three  columns  of  Table  I: 

Table    I 


e 

i 

p 

i-r 

• 

Pn 

Po 
calc. 

i 

148 
220 
320 

8 
11 
19 

700 

020 

1500 

13 
24 
72 

780 

000 

1430 

740 
962 

1382 

+  32 
62 

+  48 

410 
440 
473 

23 
26 

20 

1020 
2220 
2450 

106 
135 
168 

1810 
2085 

22S0 

1875 
2058 
2280 

35 

+  2.7 
0 

5S0 
till) 
700 

43 
56 
75 

3700 
5000 
8000 

370 

027 

1125 

3330 
4370 
6875 

3080 
3600 

4150 

+  250 

+  770 
+  2725 

The  power  consumed  by  the  motor  while  running  light 
consists  of:  The  friction  loss,  which  can  be  assumed  as  con- 
stant, a;  the  hysteresis  loss,  which  is  proportional  to  the  1.6th 
power  of  the  magnetic  flux,  and  therefore  of  the  voltage,  6e1-6; 
the  eddy  current  losses,  which  are  proportional  to  the  square 
of  the  magnetic  flux,  and  therefore  of  the  voltage,  ce2;  and  the  i2r 

The  total  power  is. 


less  in  the  windings 


p  =  a+be1-6+ce2+ri2. 


(12) 


From  the  resistance  of  the  motor  windings,  r  =  0.2  ohm, 
and  the  observed  values  of  current  i,  the  i2r  loss  is  calculated, 
and  tabulated  in  the  fourth  column  of  Table  I,  and  subtracted 
from  p,  leaving  as  the  total  mechanical  and  magnetic  losses  the 
values  of  /)()  given  in  the  fifth  column  of  the  table,  which  should 
be  expressed  by  the  equation: 


p  =  a  +  beu«+ce2. 


(13) 


This  leaves  three  constants,  a,  b,  c,  to  be  calculated. 

Plotting  now  in  Fig.  59  with  values  of  e  as  abscissas,  the 
current  i  and  the  power  p0  give  curves,  which  show  that  within 
the  voltage  range  of  the  test,  a  change  occurs  in  the  motor, 


1S-1 


EXdlXEERING   MA  THEM  A  TICS. 


as  indicated  by  the  abrupt  rise  of  current  and  of  power  beyond 
•173  volts.  This  obviously  is  due  to  beginning  magnetic  satura- 
tion of  the  iron  structure.  Since  with  beginning  saturation 
a  change  of  the  magnetic  distribution  must  be  expected,  that 
is,  an  increase  of  the  magnetic  stray  field  and  thereby  increase 
of  eddy  current  losses,  it  is  probable  that  at  this  point  the  con- 


I 

Po 

* 

.  oU 

-7000 

/( 

)  l0 

-0000 

-5000 

M 

® 

-4000 

%/ 

iU 

®  . 

-3000 

30 

^r 

Po 

-2000 

k/ 

<s 

.SO 

-1000 

e= 

Volt 

5 

10 

K 

X) 

2 

30 

a 

10 

u 

10 

a 

X) 

« 

o 

7( 

X) 

Fig.  59.     Excitation  Power  of  Induction  Motor. 

stants  in  equation  (13)  change,  and  no  set  of  constants  can  be 
expected  to  represent  the  entire  range  of  observation.  For 
the  calculation  of  the  constants  in  (13),  thus  only  the  observa- 
tions below  the  range  of  magnetic  saturation  can  safely  be  used, 
thai  is,  up  to  473  volts. 

From  equation   (13)  follows  as  the  error  of  an  individual 
observation  of  e  and  po: 

o  =  »  I  be1-*  i  <■<■--  p0; (14) 


MAXIMA   AND  MIMIMA. 


185 


hence, 

thus: 


z  =  Hd2  =  H{a+be1-6+cei—  p0f2=miuimum,      (15) 


dz 

j-  =  ?l\a+bel-«+ce2-po}=0 

dz 
db 

dz 


^=S{a+5e1-°+ce2-p!  [  :0  =  0; 

^W+Zx-^+ce--/;,  ^  =  0: 


tic 


.     (16) 


and,  if  //  is  the  number  of  observations  used  (n  =  6  in  'this 
instance,  from  e=14S  to  e  =  473),  this  gives  the  following 
equations : 

na+62e1-6+cSe2-S290=0; 
aSe1-6+6Se3.2+c2e3.6_  Sc1-«p0=0;  '■       .     .     (17) 

Substituting  in  (17)  the  numerical  values  from  Table  I  gives, 


a  + 1 1 .7  h  1 03  +  121  i  c  103  =  1 55( ) ;  J 

i 

a  +  14.IJ  b  103  +  163  c  103  =  18; 


(18) 


hence, 


and 


a +  15.1  &  L03  +  170  c  L03  =  1880; 

a  =  540; 

&  =  32.5X10-3;        s    .    .    . 

c  =  5X10-3, 

p0  =  540+0.0325e16+0.005e2.   .     . 


(19) 


(20) 


The  values  of  po,  calculated  from  equation  (20),  are  given 
in  the  sixth  column  of  Table  I,  and  their  differences  from  the 
observed  values  in  the  last  column.  As  seen,  the  errors  are  in 
both  directions  from  the  calculate!  1  values,  except  for  the  three 
highest  voltages,  in  which  the  observed  values  rapidly  increase 
beyond  the  calculated,  due  probably  to  the  appearance  of  a 


1  St  i  ENGIN  EERING   MA  Til  KM  A  TICS. 

loss  which  does  not  exist  at  lower  voltages — the  eddy  currents 
caused  by  the  magnetic  stray  field  of  saturation. 

This  rapid  divergency  of  the  observed  from  the  calculated 
values  at  high  voltages  shows  that  a  calculation  of  the  constants, 
based  on  all  observations,  would  have  led  to  wrong  values, 
and  demonstrates  the  necessity,  first,  to  critically  review  series 
of  observations,  before  using  them  for  deriving  constants,  so 
as  to  exclude  constant  errors  or  unidirectional  deviation.  It 
j mist  be  realized  that  the  method  of  least  squares  gives  the  most 
probable  value,  that  is,  the  most  accurate  results  derivable 
from  a  series  of  observations,  only  so  far  as  the  accidental 
errors  of  observations  are  concerned,  that  is,  such  errors  which 
follow  the  general  law  of  probability.  The  method  of  least 
squares,  however,  cannot  eliminate  constant  errors,  that  is. 
deviation  of  the  observations  which  have  the  tendency  to  be 
in  one  direction,  as  caused,  for  instance,  by  an  instrument  reading 
too  high,  or  too  low,  or  the  appearance  of  a  new  phenomenon 
in  a  part  of  the  observation,  as  an  additional  loss  in  above 
instance,  etc.  Against  such  constant  errors  only  a  critical 
review  and  study  of  the  method  and  the  means  of  observa- 
tion can  guard,  that  is,  judgment,  and  not  mathematical 
formalism. 

The  method  or  .east  squares  gives  the  highest  accuracy 
available  with  a  given  number  of  observations,  but  is  frequently 
very  laborious,  especially  if  a  number  of  constants  are  to  be  cal- 
culated. It,  therefore,  is  mainly  employed  where  the  number  of 
observations  is  limited  and  cannot  be  increased  at  will;  but  where 
it  can  be  increased  by  taking  some  more  observations — as  is 
generally  the  case  with  experimental  engineering  investigations 
-the  same  accuracy  is  usually  reached  in  a  shorter  time  by 
taking  a  few  more  observations  and  using  a  simpler  method 
of  calculation  of  the  constants,  as  the  ^J-method  described  in 
paragraphs  153  to  157. 


CHAPTER  V. 
METHODS  OF  APPROXIMATION. 

124.  The  investigation  even  of  apparently  simple  engineer- 
ing problems  frequently  leads  to  expressions  which  are  so 
complicated  as  to  make  the  numerical  calculations  of  a  series 
of  values  very  cumbersonme  and  almost  impossible  in  practical 
work.  Fortunately  in  many  such  cases  of  engineering  prob- 
lems, and  especially  in  the  held  of  electrical  engineering,  the 
different  quantities  which  enter  into  the  problem  are  of  very 
different  magnitude.  Many  apparently  complicated  expres- 
sions can  frequently  be  greatly  simplified,  to  such  an  extent  as 
to  permit  a  quick  calculation  of  numerical  values,  by  neglect- 
ing terms  which  are  so  small  that  their  omission  has  no  appre- 
ciable effect  on  the  accuracy  of  the  result;  that  is,  leaves  the 
result  correct  within  the  limits  of  accuracy  required  in  engineer- 
ing, which  usually,  depending  on  the  nature  of  the  problem, 
is  not  greater  than  from  0.1  per  cent  to  1  per  cent. 

Thus,  for  instance,  the  voltage  consumed  by  the  resistance 
of  an  alternating-current  transformer  is  at  full  load  current 
only  a  small  fraction  of  the  supply  voltage,  and  the  exciting 
current  of  the  transformer  is  only  a  small  fraction  of  the  full 
load  current,  and,  therefore,  the  voltage  consumed  by  the 
exciting  current  in  the  resistance  of  the  transformer  is  only 
a  small  fraction  of  a  small  fraction  of  the  supply  voltage,  hence, 
it  is  negligible  in  most  cases,  and  the  transformer  equations  are 
greatly  simplified  by  omitting  it.  The  power  loss  in  a  large 
generator  or  motor  is  a  small  fraction  of  the  input  or  output, 
the  drop  of  speed  at  load  in  an  induction  motor  or  direct- 
current  shunt  motor  is  a  small  fraction  of  the  speed,  etc.,  and 
the  square  of  this  fraction  can  in  most  cases  be  neglected,  and 
the  expression  simplified  thereby. 

Frequently,  therefore,  in  engineering  expressions  con- 
taining  small    quantities,    the    products,    squares   and    higher 

187 


|SS  ENGINEERING   MATHEMATICS. 

powers  of  such  quantities  may  be  dropped  and  the  expression 
thereby  simplified;  or,  if  the  quantities  are  not  quite  as  small 
as  to  permit  the  negleet  of  their  squares,  or  where  a  high 
accuracy  is  required,  the  first  and  second  powers  may  be  retained 
and  only  the  cubes  and  higher  powers  dropped. 

The  most  common  method  of  procedure  is,  to  resolve  the 
expression  into  an  infinite  series  of  successive  powers  of  the 
small  quantity,  and  then  retain  of  this  series  only  the  first 
term,  or  only  the  first  two  or  three  terms,  etc,  depending  on  the 
smallness  of  the  quantity  and  the  required  accuracy- 

125.  The  forms  most  frequently  used  in  the  reduction  of 
expressions  containing  small  quantities  are  multiplication  and 
division,  the  binomial  series,  the  exponential  and  the  logarithmic 
series,  the  sine  and  the  cosine  series,  etc. 

Denoting  a  small  quantity  by  s,  and  where  several  occur, 
by  Si,  s2,  S3  .  .  .  the  following  expression  holds: 

(1  ±Si)(l±S2)=l±Si±S2±SiS2, 

and,  since  x\s2  is  small  compared  with  the  small  quantities 
S]  and  s2,  or,  as  usually  expressed,  SiS2  is  a  small  quantity  of 
higher  order  (in  this  case  of  second  order),  it  may  be  neglected, 
and  the  expression  written: 

(T±si)(l±s2)  =  l±si±s2 (Is) 

This  is  one  of  the  most  useful  simplifications:  the  multiplica- 
tion of  terms  containing  small  quantities  is  replaced  by  the 
simple  addition  of  the  small  quantities. 

[f  the  small  quantities  Sj  and  s2  are  not  added  (or  subtracted) 

to  1,  but  to  other  finite;,  that  is,  not  small  quantities  a  and  b, 
a  and  l>  can  be  taken  out  as  factors,  thus, 

(a±Sl)(6±s2)  =  a6(l±^Vl±|)=a6A±^±|Y  .     (2) 

where       and  -,~  must  be  small  quantities. 
a  0 

As  seen,  in  this  case,  Sj  and  .s'L>  need  not  necessarily  be  abso- 
lutely small  quantities,  but  may  be  quite  large,  provided  that 
a  anil  h  :11c  still  larger  in  magnitude;  that  is,  S]  must  be  small 
compared  with  a,  and  s2  small  compared  with  b.    For  instance. 


METHODS  OF  APPROXIMATION.  189 

in  astronomical  calculations  the  mass  of  the  earth  (which 
absolutely  can  certainly  not  be  considered  a  small  quantity) 
is  neglected  as  small  quantity  compared  with  the  mass  of  the 
sun.  Also  in  the*  effect  of  a  lightning  stroke  on  a  primary 
distribution  circuit,  the  normal  line  voltage  of  2200  may  be 
neglected  as  small  compared  with  the  voltage  impressed  by 
lightning,  etc. 

126.  Example.  In  a  direct-current  shunt  motor,  the  im- 
pressed voltage  is  e0  =  12f)  volts;  the  armature  resistance  is 
ro=0.02  ohm;  the  field  resistance  is  ri=50  ohms:  the  power 
consumed  by  friction  is  p/=300  watts,  and  the  power  consumed 
by  iron  loss  is  /},-  =  400  watts.  What  is  the  power  output  of 
the  motor  at  io  =  50,  100  and  150  amperes  input'.' 

The  power  produced  at  the  armature  conductors  is  the 
product  of  the  voltage  e  generated  in  the  armature  conductors, 
and  the  current  i  through  the  armature,  and  the  power  output 
at  the  motor  pulley  is, 

p=ei-pf-pi (3) 

The  current  in  the  motor  field  is  — ,  and  the  armature  current 

r\ 

therefore  is, 

?=?o (4) 

/'1 

where  —  is  a  small  quantity,  compared  with  v'n. 
>'i 

The  voltage  consumed  by  the  armature  resistance  is  rni, 
and  the  voltage  generated  in  the  motor  armature  thus  is: 

e=e0— r0i, (5) 

where  tq%  is  a  small  quantity  compared  with  c,i- 
Substituting  herein  for  i  the  value  (\)  gives, 

e=e0— r0Uo— — ) (fi) 

Since  the  second  term  of  (6)  is  small  compared  with  eo, 
and  in  this  second  term,  the  second  term  is  small  com- 
pared  with  /'o,  it   can  be  neglected  as  a  small  term  of  higher 


L90 


ENGINEERING   MA  THEMA  TH  'S. 


order;    that    is,  as  small  compared  with  a    small  term,  and 
expression  (6)  simplified  to 

e  =  eo-r0in (7) 

Substituting  (4)  and  (7)  into  (3)  gives, 


p=(e0- r0io) ( /(> -~)~ Vf- Pi 

-•tov-zA1-?*)-*-*- 


(S) 


Expression  (8)  contains  a  product  of  two  terms  with  small 
quantities,  which  can  be  multiplied  by  equation  (1),  and  thereby 
gives, 


.  A     r0?*o      e0  \ 
p  =  eQiQ[l--—  --Jj-Pf-Pi 

.  .9     eo2 

=eo^o-ro^o2— —  ~Pf—pi-  ■ 


(9) 


Substituting  the  numerical  values  gives, 

p  =  125i0-0.02io2-  562.5-  300-  400 
=  125i"o-0.02i02-1260  approximately; 

thus,  for  ?;0=50,  100,  and  150  amperes;    p  =  4940,  11,040,  and 
17,040  watts  respectively. 

127.  Expressions  containing  a  small  quantity  in  the  denom- 
inator are  frequently  simplified  by  bringing  the  small  quantity 
in  the  numerator,  by  division  as  discussed  in  Chapter  Jl  para- 
graph 39,  that  is,  by  the  series, 


1 

l±x 


=  lT.r+.r-T.r:!  +  .r4TJ5  + 


(10) 


which  series,  if  x  is  a  small  quantity  s,  can  be  approximated 
by: 


l+.s 

1 
1-s 


=  l-s; 


=  l+s; 


(ID 


METHODS  OF  APPROXIMATION. 


191 


or,  where  a  greater  accuracy  is  required, 

1 


1+8 


1-s+s2: 


—     =  l+s+s2. 
1  —s 


(12) 


By  the  same  expressions  (11)  and  (12)  a  small  quantity 
contained  in  the  numerator  may  be  brought  into  the  denominator 
where  this  is  more  convenient,  thus: 


1  —  s 

l  —  s  =  -z ;  etc. 

1  +  s 


(13) 


More  generally  then,  an  expression  like  •     — ,  where  s  is 

B  a±s 

small  compared  with  a,  may  be  simplified  by  approximation  to 
the  form, 

b  h 

(14) 


a  ±  s 


o(l± 


b  (       s 
s  \      a\      a 


a 


or,  where  a  greater  exactness  is  required,  by  taking  in  the  second 
term, 


a  ±  s     a 


S        S2 


1  +  -+-  . 

a     w 


(15) 


128.  Example.  What  is  the  current  input  to  an  induction 
motor,  at  impressed  voltage  e0  and  slip  s  (given  as  fraction  ot 
synchronous  speed)  if  tq—jxq  is  the  impedance  of  the  primary 
circuit  of  the  motor,  and  r\  —  j.i'i  the  impedance  of  the  secondary 
circuit  of  the  motor  at  full  frequency,  and  the  exciting  current 
of  the  motor  is  neglected;  assuming  s  to  be  a  small  quantity; 
that  is,  the  motor  running  at  full  speed? 

Let  E  be  the  e.m.f.  generated  by  the  mutual  magnetic  flux, 
that  is,  the  magnetic  Mux  which  interlinks  with  primary  and 
with  secondary  circuit,  in  the  primary  circuit.  Since  the  fre- 
quency of  the  secondary  circuit  is  the  fraction  s  of  the  frequency 


1  92  ENGINEERING  MATHEMA Tit 'S. 

of  the  primary  circuit,  the  generated  e.m.f.  of  the  secondary 
circuit  is  sE. 

Since  x\  is  the  reactance  of  the  secondary  circuit  at  full 
frequency,  at  the  fraction  s  of  full  frequency  the  reactance 
of  the  secondary  circuit  is  %X\,  and  the  impedance  of  the  sec- 
ondary circuit  at  slip  s,  therefore,  is  ri  —  jsxi;  hence  the 
secondary  current  is, 

/--£-. 

•     rj-js.ri 

If  the  exciting  current  is  neglected,  the  primary  current 
equals  the  secondary  current  (assuming  the  secondary  of  the 
same  number  of  turns  as  the  primary,  or  reduced  to  the  same 
number  of  turns);  hence,  the  current  input  into  the  motor  is 

sE 

1  = : (16) 

•        T\  —  ]$Xi 

The  second  term  in  the  denominator  is  small  compared 
with  the  first  term,  and  the  expression  (16)  thus  can  be 
approximated  by 

sE  sE/       .sxA 

The  voltage  E  generated  in  the  primary  circuit  equals  the 
impressed  voltage  eo,  minus  the  voltage  consumed  by  the 
current  /  in  the  primary  impedance;  r0  —  j.i'a  thus  is 

E  =  eo-I(r0-jx0) (IS) 

Substituting  (17)  into  (18)  gives 

r,  sE ,        .    .  /       .  .<?.r ,  \ 

E  =  e0 (/•„      Xo)(l+j—) (19) 

In  expression  (19),  the  second  term  on  the  right-hand  side, 
which  is  the  impedance  drop  in  the  primary  circuit,  is  small 

compared  with  the  first   term  e0,  and   in  the  factor  (l+j—  J 
of  this  small  term,  the  small  term   /'  —   can  thus  be  neglected 


METHODS  OF  APPROXIMATION.  L93 

as  a  small  term  of  higher  order,  and  equation  (19)  abbreviated 
to 

^  =  eo-~(r0-jx0) (20) 

From  (20)  it  follows  that 


E  = 

and  by  (13), 


1 1 


E  =  eo{l~(r0-jxo)\. 


(21) 


Substituting  (21)  into  (17)  gives 


and  by  (1), 


M?1+£   ^M' 


!  =  —  )  1+1—--(ro-3x0) 
ri  {         >i      'i 


seo\ .       r0     .  x-o+xi 
=  —    1  —  s—  +  18—    — (22) 

If  then,   /oo^o+jV  is    the    exciting    current,  the    total 
current  input  into  the  motor  is,  approximately, 


Io=l+l 


00 

seal.,       To     .  xq+X\} 

129.  One  of  the  most  important    expressions  used  for  the 
reduction  of  small  terms  is  the  binomial  series: 

v       ,             n(n—  I)  „    n(n—  l)(n— 2) 
(1  ±x)n = 1  ± ns  +  — ^ — x  ± \i •'"'5 

n(»-lVn-2)(n-3') 

+—        -T7 -i'4±...      (24) 


If  x  is  a  small  term  s,  this  gives  the  approximation, 

(l±s)M  =  l  ±ns; (25) 


1  <  1 1  ENGIN  EERING   M  A  THEM  A  TICS. 

or,  using  the  second  term  also,  it  gives 

(l±s)»  =  l±ns+^-Ss2 (26) 

In  a  more  general  form,  this  expression  gives 

(<?  \ "         /       ns\ 
1±-)    =a»M  ±— j;  etc.      .     .     (27) 

By  the  binomial,  higher  powers  of  terms  containing  small 
quantities,  and,  assuming  n  as  a  fraction,  roots  containing 
small  quantities,  can  be  eliminated:  for  instance, 

n  —         —(  S\n        n/—(  S  \ 

Va±s=(a±s)n  =a«l  1±- )    =  Val  1± — ); 

1  i      ,lf1±iV"=l(lT- 


(a±s)n         I       s\n     a"\      a  J         a"\        a/' 

1  N_l      -1/      s\"»       1    / 

=  (a±s)    »=a    »  (l±-)       =-^t=    IT 


V(a±s)  V       a/  «?/a  \      na 

wi  to  /  o  \  ~ 


\  {a±s)™  =  (a±s)n=an[  1±-  j    =  va»»(  1±--1;  etc. 

One  of  the  most  common  uses  of  the  binomial  series  is  for 
the  elimination  of  squares  and  square  roots,  and  very  fre- 
quently it  can  be  conveniently  applied  in  mere  numerical  calcu- 
lations; as,  for  instance, 

•    (201)2  =  2002(l  1  — })2    40,000(l  +~ )  =40,400; 
29.92=302(l  -;l)2=-<K>(>(l  -  ig)  =900-6  =  894; 


1 


V99^= 10  VI -0.02  =  10(1 -0.02)  2  =  10(1 -0.01)  =  9.99; 

J_  1 1_ 

vT03~(l  +  0.03)] ""  ~1-0I5~  ;  i;}('' 


METHODS  OF  APPROXIMATION.  195 

130.  Example  1.  If  r  is  the  resistance,  x  the  reactance  of  an 
alternating-current  circuit  with  impressed  voltage  e,  the 
current  is 


Vr2+x2 


If  the  reactance  x  is  small  compared  with  the  resistance  r, 
as  is  the  case  in  an  incandescent  lamp  circuit,  then, 


2     r  {     +\r)  J 


-'-\ 


fl1_2W   j 


If  the  resistance  is  small  compared  with  the  reactance,  as 
is  the  case  in  a  reactive  coil,  then, 


(28) 


Example  2.  How  does  the  short-circuit  current  of  an 
alternator  vary  with  the  speed,  at  constant  field  excitation? 

When  an  alternator  is  short  circuited,  the  total  voltage 
generated  in  its  armature  is  consumed  by  the  resistance  and  the 
synchronous  reactance  of  the  armature. 

The  voltage  generated  in  the  armature  at  constant  field 
excitation  is  proportional  to  its  speed.  Therefore,  if  e0  is  the 
voltage  generated  in  the  armature  at  some  given  speed  So, 
for  instance,  the  rated  speed  of  the  machine,  the  voltage 
generated  at  any  other  speed  S  is 

S 


196  ENGINEERING   MATHEMA  TICS. 

o 

or,  if  for  convenience,  the  fraction  ~-  is  denoted  by  a,  then 

'->o 

a  =  tt     and    e  =  aeo, 
oo 

where  a  is  the  ratio  of  the  actual  speed,  to  that  speed  at  which 
the  generated  voltage  is  c0. 

[f  /•  is  the  resistance  of  the  alternator  armature,  Xq  the 
synchronous  reactance  at  speed  So,  the  synchronous  reactance 
at  speed  S  is  x  =  ax0,  and  the  current  at  short  circuit  then  is 

e  0£n 

i=  = (29) 

vr2+x2     vr2+a%02 

Usually  r  and  .r0  are  of  such  magnitude  that  r  consumes 
at  full  load  about  1  per  cent  or  less  of  the  generated  voltage, 
while  the  reactance  voltage  of  Xo  is  of  the  magnitude  of  from 
20  to  50  per  cent.  Thus  r  is  small  compared  with  x0,  and  if 
a  is  not  very  small,  equation  (29)  can  be  approximated  by 


ax4+(£)! 


9 


■-h-U—i  (30) 

x0         2  \axo/ 


Then  if  .r()  — 2()r,  the  following  relations  exist: 
a=        0.2  0.5  1.0  2.0 

.      Co 

i=    -X0.9688      0.005      0.99875      0.99969 
x0 

That  is,  the  short-circuit  current  of  an  alternator  is  practi- 
cally constant  independent  of  the  speed,  and  begins  to  decrease 
only  at  very  low  speeds. 

131.  Exponential  functions,  logarithms,  and  trigonometric 
functions  arc  (he  ones  frequently  met  in  electrical  engineering. 

The  exponential  function  is  defined  by  the  series, 

.        ,           ./■'-•     x3     x4     x5 
s±a;  =  l±^  +  |7f±p7+|r±|V+ (31) 


METHODS  OF  APPROXIMATION.  197 

and,  if  x  is  a  .small  quantity,  s,  the  exponential  function,  may 
be  approximated  by  the  equation, 

c±s  =  l±.v; '  (32) 

or,  by  the  more  general  equation, 

£±as  =  l±as; (33) 

and,  if  a  greater  accuracy  is  required,  the  second  term  may 
be  included,  thus, 

£±s  =  l±s+^ (34) 

and  then 

0?S2 

e±as  =  l±as+—r (35) 

The  logarithm  is  defined  by  logs  x  =    I  — ;  hence, 

dx 


hgs  (l±.r)=  ± 


Si 


±x 


Resolving  - into  a  scries,  by  (10),  and  then  integrating. 

gives 

Ioge  (l±x)=  ±  J  (lTx+x2^x*  +  .  .  .)dx 

.'•-     .r3     .r4     .i5 
=  ±-r~2±3"~4"±5"  •     •     "     (36) 

This  logarithmic  series  (36)  leads  to  the  approximation, 

loge(l±s)=±s; (37) 

or,  including  the  second  term,  it  gives 

loge  (l±s)=  ±s-s2 (38) 

and  the  more  general  expression  is,  respectively, 

log£  (a  ±  s)  =  log  a  [I  ±  *-  )  =  log  a  +  log  (l  ±~)  ^  log  a  ±^ ,      (39) 


L98 


KSCISEEIUNG    MATHEMATICS 


and,  more  accurately, 


S       S* 

loge(a±s)  =  loga±--^. 


(40) 


Si?i<r  logio  A^=logio  eXloge  2V=0.4343  log£  AT,  equations  (39) 

and  (40)  may  be  written  thus, 


logio(l±s)=±0.4343s; 
logio  (a±s)  =  logio  a ±0.4343  - 


(41) 


132.  The   trigonometric   functions    are   represented   by  the 

infinite  scries : 

.     x2    x*    x6 
cos  x=l  — p^+rr  — j-r+.  .  .  ; 


Eli  I! 

X*      X5      X7 


•  • 


(42) 


bill  Us        X/        lo  ~i    •  —         I—      i- •   •   . 

V 
which  wlien  s  is  a  small  quantity,  may  be  approximated  by 

coss  =  l     and     sin  s  =  s;        .     .     .     .     (43) 
or,  they  may  be  represented  in  closer  approximation  by 


cos  s  =  l 


sin  s  =  s 


2' 


6/' 


or,  by  the  more  general  expressions, 

cos  as  =  1      and     cos  as  =  1  - 

sin  as  =  as    and     sin  as  =  as 


a2s2 
2   ' 

rt3.S3 


6 


(44) 


(45) 


133.  Other  functions  containing  small  terms  may  frequently 

be    approximated    by    Taylor's    series,    or    its    special    case, 
MacLaurin's  series. 

MacLaurin's  scries  is  written  thus: 


x2. 


f\  x)  =/(0)  +xf'(0)  +p/"(0)  +  ^f'"(Q)  +. 


(46) 


METHODS  OF  APPROXIMATION. 


199 


where/',/",/"',  etc.,  arc  respectively  the  first,  second,  third, 
etc.,  differential  quotient  of/;  hence, 

/(s)=/(0)+s/'(0); 


/M=/(0)+as/'(0).   . 
Taylor's  series  is  written  thus, 

/(&+■*)  =/(&)  +x/'(6)  +j-!y/,,(6^  +t^/'"(&)  +.  . .  , 


(47^ 


.r 


(48) 


and  leads  to  the  approximations: 
f(b±s)=f{b)±sf'(b);      } 


f(b±as)=f(b)±asf'(b 


f 


(49} 


Many  of  the   previously  discussed  approximations  can   be 

considered  as  special  cases  of  (47)  and  (49). 

134.  As  seen  in  the  preceding,  convenient  equations  for  the 
approximation  of  expressions  containing  small  terms  are 
derived  from  various  infinite  series,  which  are  summarized 
below : 

—  =  lTx+x2fx3+x!T.  .  .  ; 

v       -,             11  11 -D    ,     n{n—  l)(n—  2)  _ 
(I±s)"  =  l±na;+-Lj2 x2±~       ~d -r5  +  - 

X2      X3      X4 


e±*  =  1  ±x+—±—+—±i 

x2     x3     .r4 
logs  (l±x)=  ±x-  —  ±"o-  T±. 


x2     x4     .r6 


1-    (50) 


COS  X  =  1  —  —  -  ■  77-  -  -  777  +  .  .  . 

-  0 


x3     .r5     .r7 

sin  x  =  x  —  nr  +tv  —  7=-  4- . 
3      0       / 


Ax)  =/(0)  +x/'(0)  +|/"(0)  +|3/,/,(0)  +. 


x- 


.r5 


/(& ±x)  =/(&)  ±xf  (6)  +  ny/"(5)  ±t^/,,,(&)  + 


200 


ENGINEERING   MA  THEMATICS. 


The  first  approximations,  derived  by  neglecting  all  higher 
terms  but  the  first  power  of  the  small  quantity  x=s  in  these 
series,  are: 

1 


TTs=lTs'> 

(l±.sV'=--l  ±ns; 

[  +  -]: 

a' a  —  1 ) 
L+       2 

s--' 

£±s  =  l  ±S; 

[4 

o%£  1 1  ±s)  -  ±.s: 

[-a= 

cos  s     1 ; 

[-ft 

sin  s=s; 

[-a 

/(«)  =/(0)  + 

f{0); 

[+^/"(0) 

; 

f(b±s)=f(b)±. 

if{b); 

[+|W~>) 

• 

(51) 


and,  in  addition  hereto  is  to  be  remembered  the  multiplication 


•(lie. 


(1-  ±Si)(l  ±S2)=l±Si  ±S2)         [±SiS2]. 


1 52 ) 


135.  The  accuracy  of  the  approximation  can  be  estimated 
by  calculating  the  next  term  beyond  that  which  is  used. 
This  term  is  given  in  brackets  in  the  above  equations  (50) 
and  (51). 

Thus,  when  calculating  a  series  of  numerical  values  by 
approximation,  for  the  one  value,  for  which,  as  seen  by  the 
nature  of  the  problem,  the  approximation  is  least  close,  the 
next  term  is  calculated,  and  if  this  is  less  than  the  permissible 
limits  of  accuracy,  the  approximation  is  satisfactory. 

For  instance,  in  Example  2  of*  paragraph  130,  the  approxi- 
mate value  of  the  short-circuit  current  was  found  in  (30),  as 

21 


.-ro  [        2  \axo/    \ 


METHODS  OF  APPROXIMATION.  20] 

The  next  term  in  the  parenthesis  of  equation  (30),  by  the 

7i  (n  —  1 ) 
binomial,  would  have  been   -\ — — s2;    substituting  n=—\\ 

s=l  — )     the  next  becomes  +'o\ —  )  .     The  smaller  the  a,  the 
\a.To/  8  \axo/ 

less  exact  is  the  approximation. 

The  smallest  value  of  a,  considered  in  paragraph  130,  was 

a=0.2.     For  z0  =  20r,   this   gives    +~(-   -)    =0.00146,   as  the 

8  \axo/ 

value  of  the  first  neglected  term,  and  in  the  accuracy  of  the 
result  this  is  of  the  magnitude  of  —  X 0.00146,  out  of -- X  0.9688, 

Xq  Xq 

the  value  given  in  paragraph  130;    that  is,  the  approximation 

•  ,  •    0.00146 
gives  the  result  correctly  within   .. ,.,,.,,.  =0.0015  or  withm  one- 

sixth  of  one  per  cent,  which  is  sufficiently  close  for  all  engineer- 
ing purposes,  and  with  larger  a  the  values  are  still  closer 
approximations. 

136.  It  is  interesting  to  note  the  different  expressions, 
which  are  approximated  by  (1+s)  and  by  (1  —  s).  Some  of 
them  are  given  in  the  following: 

1+8=  1-8= 

1  1 


1-S'  1+.S 


i+-V;  (i-'-Y; 

nl  \      n  / 


s\2  I      sV 


1 


1+1 

n, 


m 

x-ffs 

n  —  m 
n2 


y 


202 


E  VGINEERING   MATHEMATICS. 


\  1  +2s; 
1 


Vl-2s 


/r+i 


t_ 

y/l  —  ns 


4 


1  +  ms 


1  —  (n—m)s' 
etc. 


2- 


\l+s 


vl  —  ns; 
1 


VT+ns 
[ 


ns 

-t. 


<! 


1  —  ms 


L  +  (»— Wl)s' 

etc. 


2-£s: 


l+loge  (1+s); 

l+loge(l-s); 

l-loge(l-s); 

1- loge  (1+s); 

l+nloge(l+^); 

l+nlog,(l— j; 

l-»log,U-0; 

l-"log£(l+~); 

i+^Vl^I5 

1+Iog£Vi+s; 

1     1(,Nl+s; 

i  ^Vi-S; 

etc. 

etc. 

1  +sin  s; 

1  — sin  6-; 

■    s 
1  +nsm  -; 
n' 

1  —  n  sin  — : 

METHODS  OF  APPROXIMATION. 


203 


,      1     . 

1-f  —  sm  ns; 
n 


cos  v— 2s; 

etc. 


1 —  sin  ns; 

n 

cos  vzs; 
etc. 


137.  As  an  example  may  be  considered  the  reduction  to  its 
simplest  form,  of  the  expression: 


\  a  \(tt+si,)3{4—  sin  Qs2\  -\oua  cos2 


e-3s\a+2si)   1  — alog< 


a  —  So  1 


a+so  J 


\  v'a—  2si 


then, 


,  1/  Si\3/4  1/  S   Si' 


4  —  sin  6s2  =  4  M  —  j  sin  Gs2  j  =  4  ( 1  — '-  s2  J • 


*S  Si 

ea  =1+2-; 
a 


cos- 


Sl\2  Si- 

1--     =1-2-; 
a/  a 


r-3sS  =  ^_3g    • 


a+2s1  =  a(l+2-J; 


;r^       . 

/  (l  1  /  S2 

a 


=  1— a  log£(  L— — )  =      f  s2; 


\  a-2si  =  a1/2  1 


2si\1/2  Si 

=  aV2  l- 


a 


a 


•_>(  M  EM  UN  EERING   MA  Til  KM  A  TJ(  'S. 

hence. 


F  = 


CTl/2xa3/4(1+|  ll)  X4(l-|.,>)  Xa^x(l  +2^)(l-2Ji) 

(l-3.s,)Xa(l+2^j(1+.^)Xa1/2(l-^j 


1+|£l_|S2+2£i_2£i 

4  a      2  a       a 

I  Si  Si 

a3/2   1_3,s  +2-i+.s2 

\  a  a . 

138.  As  further  example  may  be  considered  the  equations 
of  an  alternating-current  electric  circuit,  containing  distributed 
resistance,  inductance,  capacity,  and  shunted  conductance,  for 
instance,  a  long-distance  transmission  line  or  an  underground 
high-potential  cable. 

Equations  of  the  Transmission  Line. 

Let  I  be  the  distance  along  the  line,  from  some  starting 
point;  E,  the  voltage;  /,  the  current  at  point  /,  expressed  as 
vector  quantities  or  general  numbers;  Zq^tq  —  jxq,  the  line 
impedance  per  unit  length  (for  instance,  per  mile);  Yo=-g0  —  jlm 
=  line  admittance,  shunted,  per  unit  length;  that  is,  r()  is  the 
ohmic  effective  resistance;  .r0,  the  self-inductive  reactance; 
60,  the  condensive  susceptance,  that  is,  wattless  charging 
current  divided  by  volts,  and  </o  =  energy  component  of  admit- 
tance, that  is,  energy  component  of  charging  current,  divided 
by  volts,  per  unit  length,  as,  per  mile. 

Considering  a  line  element  dl,  the  voltage,  dE,  consumed 
by  the  impedance  is  Z^hll,  and  the  current,  <II,  consumed  by 
the  admittance  is  )'0E'll;  hence,  the  following  relations  may  !»■ 
written : 

dE 

TB-SoT; CD 

f=y°? » 


METHODS  OF   APPROXIMATION. 


205 


Differentiating  (1),  and  substituting  (2)  therein  gives 


d2E 
-jp^ZoYoE, 


and  from  (1)  it  follows  that, 


7  = 


1    dE 


Zq  dl 
Equation  (3)  is  integrated  by 

E  =  AeBl, 
and  (5)  substituted  in  (3)  gives 


B=±VZ0Yo)     .     . 
hence,  from  (5)  and  (4),  it  follows 


% 


|A1£+x/ZoFo'—  A2£-y^ZoYo1}- 


(3) 


(4) 


(5) 


•  (0) 

•  (7) 

•  (8) 


Next  assume 

l  =  lc.,      the  entire  length  of  line; 
Z  =  IqZ0,  the  total  line  impedance ;      !-,...     (9) 
and  Y  =  I0Y0,  the  total  line  admittance; 

then,  substituting  (9)  into  (7)  and  (8),  the  following  expressions 
are  obtained : 


Y 


!i~JV{AlS+^-A2e-^}, 


]Z 


(10) 


as  the  voltage  and  current  at  the  generator  end  of  the  line. 

139.  If  now  E{)  and  /(»  respectively  are  the  current  and 
voltage  at  the  step-down  end  of  the  line,  for  1  =  0,  by  sub- 
stituting 1  =  0  into  (7)  and  (8), 

A,+A,  =  E0; 


A1-A2  =  I( 


(11) 


206 


ENGINEERING   MATHEMA  Tit 'S. 


e±ZY  =  l±VZY 


ZY    ZY\  ZY     Z-Y1    Z2Y2VZY 


Substituting  in  (10)  for  the  exponential  function,  the  scries, 

+  .  .  . 
(12) 


±- 


"aT 


1-0 


ZY    Z2Y2\       /==/      ZY 


21 


(> 


120 


and  arranging  by   (Ai+A2)  and    (Ai— 42),  and  substituting 
herefor  the  expressions  (11),  gives 


ZY    Z2Y2}      r7T     \       ZY    Z2Y2} 

?i-?o|l+-2-+-^-}+^/o  {1.+-6-+I26-J 

T      r    f       ZY    Z2Y2)      TrT1  f       ZY    Z2Y2} 


24 


120 


(13) 


When  1=—Iq,  that  is,  for  Eq and  70 at  the  generator  side,  and 
Ei  and  h  at  the  step-down  side  of  the  line,  the  sign  of  the 
second  term  of  equations  (13)  merely  reverses. 

140.  From  the  foregoing,  it  follows  that,  if  Z  is  the  total 
impedance;  Y,  the  total  shunted  admittance  of  a  transmission 
line,  'E0  and  70,  the  voltage  and  current  at  one  end;  Ex  and  I\, 
the  voltage  and  current  at  the  other  end  of  the  transmission 
line;  then, 


w     __  r     zy  Z2Y2}    __  r     zy  ^r*]    i 


(i  =  (o     l+^7-+-^T-    ±YEo{  l+^r-  + 
I        -       -4  J        'I 


-,     (14) 


(> 


120 


where  the  plus  sign  applies  if  7?0,  70  is  the  step-down  end, 
the  minus  sign,  if  E0,  70  is  the  step-up  end  of  the  transmission 
line. 

In  practically  all  cases,  the  quadratic  term  can  be  neglected, 
and  the  equations  simplified,  thus, 


B,  K„jn4n±z/.>{w-~! 


-    I 


1 


6 
ZY 


h-!o{l  +  ™}±YE0{l+-f. 


(15) 


z2t 


and  the  error  made  hereby  is  of  the  magnitude  of  less  than  \' 


MET  lions  OF  APPROXIMATION.  207 

Except  in  the  case  of  very  long  lines,  the  second    term  of 
the  second  term  can  also  usually  be  neglected,  which  givet= 


£\  =  #o(l+-^)±Z/(1: 


.    .     (16) 


and  the  error  made  hereby  is  of  the  magnitude  of  less  than  — 

6 

of  the  line  impedance  voltage  and  line  charging  current. 

141.  Example.  Assume  200  miles  of  60-cycle  line,  on  non- 
inductive  load  of  eu  =  100,000  volts;  and  /0  =  100  amperes. 
The  line  constants,  as  taken  from  tables  are  Z  =  104  — 140/  ohms 
and  Y  =  —0.0013/  ohms;   hence, 

ZY=-  (0.182  +0.1367); 

Si  =  100000(1 -0.091 -0.088/)  +100(104-104/) 
=  101400-20800/,  in  volts; 

Ix  =100(1-0.091  -  0.068/)- 0.0013/ X 100000 
=  91  —  136.8/,  in  amperes. 

^  .    m     0.174X0.0013     0.226 

The  error  is  -+  =  -        —     —  =  —r-  =•  0.038. 
(>  b  b 

In  Elf  the  neglect  of  the  second  term  of  e/0  =  17,400,  gives 
an  error  of  0.038  X  17,400  =  000  volts  =  0.6  per  cent. 

In  I1}  the  neglect  of  the  second  term  of  yE0  =  130,  gives  an 
error  of  0.038x130  =  5. amperes  =  3  per  cent. 

Although  the  charging  current  of  the  line  is  130  per  cent 
of  output  current,  the  error  in  the  current  is  only  3  per  cent. 

Using  the  equations  (15),  which  are  nearly  as  simple,  brings 

zhj2    0.22C>- 
the  error  down  to  ~rr^~ 7m — =0.0021,  or  less  than  one-quarter 

24  24 

per  cent. 

Hence,  only  in  extreme  cas<\s  the  equations  (14)  need  to  be 

z4y4 
used.     Their  error  would  be  less  than  -=-r-r  =  3.0XlO~G,  or  one 


720     -— "     > 


three-thousandth  per  cent. 


208  ENGINEERING   MATHEMATICS. 

The-  accuracy  of  the  preceding  approximation  can  be  esti- 
mated by  considering  the  physical  meaning  of  Z  and  Y:  Z 
is  the  line  impedance;    hence  ZI  the  impedance  voltage,  and 

zi 

u=    ...  the  impedance  voltage  of  the  line,  as  fraction  of  total 

voltage;    Y  is  the  shunted  admittance:   hence  YE  the  charging 

YE 

current,  and  *,  =  _y~>  the  charging  current  of  the  line,  as  fraction 

of  total  current. 

Multiplying  gives  uv=ZY;  that  is,  the  constant  ZY  is  the 
product  of  impedance  voltage  and  charging  current,  expressed 
as  fractions  of  full  voltage  and  full  current,  respectively.  In 
any  economically  feasible  power  transmission,  irrespective  of 
its  length,  both  of  these  fractions,  and  especially  the  first, 
must  be  relatively  small,  and  their  product  therefore  is  a  small 
quantity,  and  its  higher  powers  negligible. 

In  any  economically  feasible  constant  potential  transmission 
line  the  preceding  approximations  are  therefore  permissible. 


CHAPTER    VI. 
EMPIRICAL  CURVES. 

A.  General. 

142.  The  results  of  observation  or  tests  usually  are  plotted 

in  a  curve.     Such  curves,  for  instance,  are  given  by  the  core 

loss  of  an  electric  generator,  as  function  of  the  voltage;    or, 

the  current  in  a  circuit,  as  function  of  the  time,  etc.     When 

plotting  from  numerical  observations,  the  curves  are  empirical, 

and  the  first  and  most    important    problem  which  has  to  be 

solved  to  make  such  curves  useful  is  to  find  equations  for  the 

same,  that  is,  find   a   function,    y=f(x),   which   represents   the 

curve.     As  long  as  the  equation  of  the  curve  is  not  known  its 

utility  is  very  limited.     While  numerical  values  can  be  taken 

from  the  plotted  curve,  no  general  conclusions  can  be  derived 

from  it,  no  general  investigations  based  on  it  regarding  the 

conditions  of  efficiency,  output,  etc.     An  illustration  hereof  is 

afforded  by  the  comparison  of  the  electric  and  the  magnetic 

circuit.     In  the  electric  circuit,  the  relation  between  e.m.f.  and 

e 
current  is  given  by  Ohm's  law,  %=—,  and  calculations  are  uni- 

r 

versally  and  easily  made.  In  the  magnetic  circuit,  however, 
the  term  corresponding  to  the  resistance,  the  reluctance,  is  not 
a  constant,  and  the  relation  between  m.m.f.  and  magnetic  flux 
cannot  be  expressed  by  a  general  law,  but  only  by  an  empirical 
curve,  the  magnetic  characteristic,  and  as  the  result,  calcula- 
tions of  magnetic  circuits  cannot  be  made  as  conveniently  and 
as  general  in  nature  as  calculations  of  electric  circuits. 

If  by  observation  or  test  a  number  of  corresponding  values 
of  the  independent  variable  x  and  the  dependent  variable  y  are 
determined,  the  problem  is  to  find  an  equation,  y=fU),  which 
represents  these  corresponding  values:  X\,  X2,  X3  .  .  .  xn,  and 
Vh  2/2;  2/3  ••  •  Vn,  approximately,  that  is,  within  the  errors  of 
observation. 

209 


210  ENGINEERING   MATHEMATICS. 

The  mathematical  expression  which  represents  an  empirical 
curve  may  be  a  rational  equation  or  an  empirical  equation. 
It  is  a  rational  equation  if  it  can  be  derived  theoretically  as  a 
conclusion  from  some  general  law  of  nature,  or  as  an  approxima- 
tion thereof,  but  it  is  an  empirical  equation  if  no  theoretical 
reason  can  be  seen  for  the  particular  form  of  the  equation. 
For  instance,  when  representing  the  dying  out  of  an  electrical 
current  -in  an  inductive  circuit  by  an  exponential  function  of 
time,  we  have  a  rational  equation:  the  induced  voltage,  and 
therefore,  by  Ohm's  law,  the  current,  varies  proportionally  to  the 
rate  of  change  of  the  current,  that  is,  its  differential  quotient, 
and  as  the  exponential  function  has  the  characteristic  of  being 
proportional  to  its  differential  quotient,  the  exponential  function 
thus  rationally  represents  the  dying  out  of  the  current  in  an 
inductive  circuit.  On  the  other  hand,  the  relation  between  the 
loss  by  magnetic  hysteresis  and  the  magnetic  density:  W=  rj(S>l-G, 
is  an  empirical  equation  since  no  reason  can  be  seen  for  this 
law  of  the  1.6th  power,  except  that  it  agrees  with  the  observa- 
tions. 

A  rational  equation,  as  a  deduction  from  a  general  law  of 
nature,  applies  universally,  within  the  range  of  the  observa- 
tions as  well  as  beyond  it,  while  an  empirical  equation  can  with 
certainty  be  relied  upon  only  within  the  range  of  observation 
from  which  it  i<  derived,  and  extrapolation  beyond  this  range 
becomes  increasingly  uncertain.  A  rational  equation  there- 
fore is  far  ]  (referable  to  an  empirical  one.  As  regards  the 
accuracy  of  representing  the  observations,  no  material  difference 
exists  between  a  rational  and  an  empirical  equation.  An 
empirical  equation  frequently  represents  the  observations  with 
great  accuracy,  while  inversely  a  rational  equation  usually 
doe-  not  rigidly  represent  the  observations,  for  the  reason  that 
in  nature  the  conditions  on  which  the  rational  law  is  based  are 
rarely  perfectly  fulfilled.  For  instance,  the  representation  of  a 
decaying  current  by  an  exponential  function  is  based  on  the 
assumption  that  the  resistance  and  the  inductance  of  the  circuit 
are  constant,  and  capacity  absent,  and  none  of  these  conditions 
can  ever  be  perfectly  satisfied,  and  thus  a  deviation  occurs  from 
the  theoretical  condition,  by  what  is  called  "  secondary  effects." 

i43-  To  derive  an  equation,  which  represents  an  empirical 
curve,  careful  consideration  should  firsl  be  given  to  the  physical 


EMPIRICAL  CURVES.  211 

nature  of  the  phenomenon  which  is  to  be  expressed,  since 
thereby  the  number  of  expressions  which  may  be  tried  on  the 
empirical  curve  is  often  greatly  reduced.  Much  assistance  is 
usually  given  by  considering  the  zero  points  of  the  curve  and 
the  points  at  infinity.  For  instance,  if  the  observations  repre- 
sent the  core  loss  of  a  transformer  or  electric  generator,  the 
curve  must  go  through  the  origin,  that  is,  y  =  0  for  x=0,  and 
the  mathematical  expression  of  the  curve  y=f(x)  can  contain 
no  constant  term.  Furthermore,  in  tins  case,  with  increasing  x, 
y  must  continuously  increase,  so  that  for  x  =  00 ,  y=<x>.  Again, 
if  the  observations  represent  the  dying  out  of  a  current  as 
function  of  the  time,  it  is  obvious  that  for  x=<x>,  y=0.  In 
representing  the  power  consumed  by  a  motor  when  running 
without  load,  as  function  of  the  voltage,  for  x  =  0,  y  cannot  be 
=  0,  but  must  equal  the  mechanical  friction,  and  an  expression 
like  y=Axa  cannot  represent  the  observations,  but  the  equation 
must  contain  a  constant  term. 

Thus,  first,  from  the  nature  of  the  phenomenon,  which  is 
represented  by  the  empirical  curve,  it  is  determined 

(a)  Whether  the  curve  is  periodic  or  non-periodic. 

(b)  Whether  the  equation  contains  constant  terms,  that  is, 
for  x=0,  y^O,  and  inversely,  or  whether  the  curve  passes 
through  the  origin:  that  is,  y  =  0  for  x-=0,  or  whether  it  is 
hyperbolic;  that  is,  y=  gc  for  x  =  0,  or  .r=oc  for  y  =  0. 

(c)  What  values  the  expression  reaches  for  00.  That  is, 
whether  for  x=  go,  y  =  zc,  or  y  =  0,  and  inversely. 

(d)  Whether  the  curve  continuously  increases  or  decreases,  or 
reaches  maxima  and  minima. 

(e)  Whether  the  law  of  the  curve  may  change  within  the 
range  of  the  observations,  by  some  phenomenon  appearing  in 
some  observations  which  does  not  occur  in  the  other.  Thus, 
for  instance,  in  observations  in  which  the  magnetic  density 
enters,  as  core  loss,  excitation  curve,  etc.,  frequently  the  curve 
law  changes  with  the  beginning  of  magnetic  saturation,  and  in 
this  case  only  the  data  below  magnetic  saturation  would  be  used 
for  deriving  the  theoretical  equations,  and  the  effect  of  magnetic 
saturation  treated  as  secondary  phenomenon.  Or,  for  instance, 
when  studying  the  excitation  current  of  an  induction  motor, 
that  is,  the  current  consumed  when  running  light,  at  low- 
voltage  the  current  mav  increase  again  with  decreasing  voltage. 


212  ENGIN  EERING   M.  I  Til  KM  A  TICS. 

instead  of  decreasing,  as  result  of  the  friction  load,  when  the 
voltage  is  so  low  that  the  mechanical  friction  constitutes  an 
appreciable  part  of  the  motor  output.  Thus,  empirical  curves 
can  he  represented  by  a  single  equation  only  when  the  physical 
conditions  remain  constant  within  the  range  of  the  observations. 

From  the  shape  of  the  curve  then  frequently,  with  some 
experience,  a  guess  can  be  made  on  the  probable  form  of  the 
equation  which  may  express  it.  In  this  connection,  therefore, 
it  is  of  the  greatest  assistance  to  be  familiar  with  the  shapes  of 
the  more  common  forms  of  curves,  by  plotting  and  studying 
various  forms  of  equations  y=f(x). 

By  changing  the  scale  in  which  observations  are  plotted 
the  apparent  shape  of  the  curve  may  be  modified,  and  it  is 
therefore  desirable  in  plotting  to  use  such  a  scale  that  the 
average  slope  of  the  curve  is  about  45  deg.  A  much  greater  or 
much  lesser  slope  should  lie  avoided,  since  it  does  not  show  the 
character  of  the  curve  as  well. 

B.  Non-Periodic  Curves. 

144-  The  most  common  non-periodic  curves  are  the  potential 
series,  the  parabolic  and  hyperbolic  curves,  and  the  exponential 
and  logarithmic  curves. 


'to1- 


The  Potential  Series. 

Theoretically,  any  set  of  observations  can  be  represented 
exactly  by  a  potential  series  of  any  one  of  the  following  forms: 

y  =  (io  +  alx+a2x2+a3x:i-{  .  .  .  ■       ....     (1) 

y  =  aix+a2x2+aix3  +  .  .  .  ; '   (2) 

(i\     «•>     a- 

^ao+_+_ +_+.... (3) 

<i\     <t>    as 

y=-^+^+^+ <" 

if  a  sufficient  ly  large  number  of  terms  are  chosen. 

For  instance,  if  //  corresponding  numerical  values  of  .rand  y 
are  given,  X\,  //,;    x2}  y2\    ■  .  .  Xn,  //„,  they  can  he  represented 


EMI'Ih'KAL  CURVES. 


213 


by  the  scries  (1),  when  choosing  as  many  terms  as  required  to 

give  n  constants  a: 


y  =  a0  +  axx  +  a2x2  + .  .  .+an_inn~l. 


(5) 


By  substituting  the  corresponding  values  X\,  y\)  x2,  y%, .  .  . 
into  equation  (5),  there  are  obtained  n  equations,  which  de- 
termine the  n  constants  (to,  a,\,  a2,  .  .  .  an_\. 

Usually,  however,  such  representation  is  irrational,  and 
therefore  meaningless  and  useless. 

Table  I. 


e 

-0.5 

+  2x 

+  2.5.r? 

-1.5.r3 

+  1.5.fi 

-2a:6 

+  j6 

0.4 
0.6 
O.S 

0.63 
1.36 

2.18 

-0.5 
-0.5 
-0.5 

+  0.8 
+  1.2 
+  1.6 

+  0.4 
+  0.9 
+  1.6 

-0.10 
-0.32 
-0.77 

+  0.04 
+0.19 

+  0.61 

-  0.02 

-  0.16 

-  0.65 

0 
+   0.05 
+   0.26 

1.0 

1.2 
1.4 

3.00 
3.03 
6 .  22 

-0.5 
-0.5 
-0.5 

+  2.0 
+  2.4 
+  2.8 

+  2.5 
+  3.6 
+  4.9 

-1.50 
-2.59 
-4.12 

+  1.50 
+  3.11 
+  5.76 

-2.00 

-    4.98 
-10.76 

+    1.00 
+   2.89 
+   6.13 

16 

S .  59 

-0.5 

+  3.2 

+  6.4 

-6.14 

+  9.83 

-20.97 

+  16.78 

Let,  for  instance,  the  first  column  of  Table  I  represent  the 
voltage,  Ym=X}  m  nundreds  of  volts,  and  the  second  column 

the  core  loss,  p,  =  y,  in  kilowatts,  of  an  125- volt  100-h.p.  direct- 
current  motor.  Since  seven  sets  of  observations  are  given, 
they  can  be  represented  by  a  potential  series  with  seven  con- 
stants, thus, 

y  =  an +ci\i  +a-2x2+.  .  .  +a6.r6,     ....     ((5) 

and  by  substituting  the  observations  in  (6),  and  calculating  the 
constants  a  from  the  seven  equations  derived  in  this  manner, 
there  is  obtained  as  empirical  expression  of  the  core  loss  of 
the  motor  the  equation, 


y  =  _  0. 5  +  2x  +  2 .  5x2  -  1 .  o.r3  + 1 .  o.r4  -  2x5  +  x6. 


(7) 


This  expression  (7),  however,  while  exactly  representing 
the  seven  observations,  has  no  physical  meaning,  as  easily 
seen  by  plotting  the  individual  terms.     In  Fig.  (10,  y  appears 


214 


ENGINEERING   MATHEMATICS. 


as  the  resultant  of  a  number  of  large  positive  and  negative 
terms.  Furthermore,  if  one  of  the  observations  is  omitted, 
and  the  potential  series  calculated  from  the  remaining  six 
values,  a  series  reaching  up  to  .r5  would  be  the  result,  thus, 

y  =  a0+aix+a2x2  +  . .  ,+a5x5,     ....     (8) 


16 

12 

8 

i 

>li> 

3D* 

]L- 

£=^ 

*t& 

'0 

-0.5 

-4 

^*, 

-8 

V* 

f 

-12 

-16 

-30 

x  = 

0 

2 

0 

4 

0 

6 

0 

8 

lio 

1 

2 

1 

1 

1 

Fig.  60.     Terms  of  Empirical  Expression  of  Excitation  Power. 

but  the  constants  a  in  (8)  would  have  entirely  different  numer- 
ical values  from  those  in  (7),  thus  showing  that  the  equation 
(7)  has  no  rational  meaning. 

145-  The  potential  series  (1)  to  (4)  thus  can  be  used  to 
represent  an  empirical  curve  only  under  the  following  condi- 
t  ions: 

1.  If  the  successive  coefficients  a(),  ah  a2)  .  .  .  decrease  in 
value  so  rapidly  that  within  the  range  of  observation  the 
higher  terms  become  rapidly  smaller  and  appear  as  mere 
secondary  terms. 


EMPIRICAL  CURVES. 


215 


2.  If  the  successive  coefficients  a  follow  a  definite  law, 
indicating  a  convergent  series  which  represents  some  other 
function,  as  an  exponential,  trigonometric,  etc. 

3.  If  all  the  coefficients,  a,  are  very  small,  with  the  exception 
of  a  few  of  them,  and  only  the  latter  ones  thus  need  to  be  con- 
sidered. 

Table  II. 


• 

X 

u 

vl 

i/i 

0.4 

O.S!) 

0 .  88 

0.01 

0.6 

1.35 

1.34 

0.01 

o.s 

1 .  96 

1.94 

0 .  02 

1.0 

2.72    ' 

2.70 

0.02 

1.2 

3.62 

2.59 

0.03 

1.4 

4 .  63 

4.59 

0.01 

1.6 

5.76 

5.65 

0.11 

For  instance,  let  the  numbers  in  column  1  of  Table  II 
represent  the  speed  x  of  a  fan  motor,  as  fraction  of  the  rated 
speed,  and  those  in  column  2  represent  the  torque  y,  that  is, 
the  turning  moment  of  the  motor.  These  values  can  be 
represented  by  the  equation, 

y  =  0.5  +0.02z  +  2.5.r2  -  0.3x*  +  0.01  o.r*  -  0.02.;. 5  +  O.Olx6.     (9) 

In  this  case,  only  the  constant  term  and  the  terms  with 
x2  and  x3  have  appreciable  values,  and  the  remaining  terms 
probably  are  merely  the  result  of  errors  of  observations,  that  is, 
the  approximate  equation  is  of  the  form, 

y  =  cio  +  a-2x2  +  o.zx3 (10) 

Using  the  values  of  the  coefficients  from  (9),  gives 

t/  =  0.5+2.5x2-0.3x3 (11) 

The  numerical  values  calculate. I  from  (11)  are  given  in  column 
3  of  Table  II  as  if,  and  the  difference  between  them  and  the 
observations  of  column  2  are  given  in  column  4,  as  //i. 


216  ENGINEERING   MA  THEMATIC 'S. 

The  values  of  column  4  can  now  be  represented  by  the  same 
form  of  equation,  namely, 

yi=b0+b2x2+bsx3, (12) 

in  which  the  constants  &o,  &2,  &3  are  calculated  by  the  method 
of  least  squares,  as  described  in  paragraph  120  of  Chapter  IV, 
and  give 

2/i =0.031 -0.093^2+ 0.076a:3 (13) 

Equation  (13)  added  to  (11)  gives  the  final  approximate 
equation  of  the  torque,  as, 

2/o=0.531  +2.407x2-  0.224x3 (14) 

The  equation  (14)  probably  is  the  approximation  of  a 
rational  equation,  ^ince  the  first  term,  0.531,  represents  the 
bearing  friction;  the  second  term,  2.407a;2  (which  is  the  largest), 
represents  the  work  done  by  the  fan  in  moving  the  air,  a 
resistance  proportional  to  the  square  of  the  speed,  and  the 
third  term  approximates  the  decrease  of  the  air  resistance  due 
to  the  churning  motion  of  the  air  created  by  the  fan. 

In  general,  the  potential  series  is  of  limited  usefulness;  it 
rarely  has  a  rational  meaning  and  is  mainly  used,  where  the 
curve  approximately  follows  a  simple  law,  as  a  straight  line, 
to  represent  by  small  terms  the  deviation  from  this  simple  law, 
that  is,  the  secondary  effects,  etc.  Its  use,  thus,  is  often 
temporary,  giving  an  empirical  approximation  pending  the 
derivation  of  a  more  rational  law. 

The  Parabolic  and  the  Hyperbolic  Curves. 

146.  One  of  the  most  useful  classes  of  curves  in  engineering 
are  those  represented  by  the  equation, 

y  =  ax"; (15) 

or,  the  more  general  equation, 

y—b  =  a{x—c)n (16) 

Equation  (Hi)  differs  from  (15)  only  by  the  constant  terms  b 
and  c;    that  is,  it  gives  a  different   location  to  the  coordinate 


EMPIRICAL  CURVES. 


217 


(enter,  but  the  curve  shape  is  the  same,  so  that  in  discussing 
die  general  shapes,  only  equation  (15)  need  be  considered. 

If  n  is  positive,  the  curves  y  =  axn  are  parabolic  curves, 
j  assing  through  the  origin  and  increasing  with  increasing  .r. 
If  n>l,  y  increases  with  increasing  rapidity,  if  n<l,  y  increases 
with  decreasing  rapidity. 

If  the  exponent  is  negative,  the  curves  y  =  ax~n  = —  are 

hyperbolic  curves,  starting  from  w=oo  for  £=0,  and  decreasing 
to  y  =  0  for  x  =  oo. 

n=l  gives  the  straight  line  through  the  origin,  n=0  and 
?i=co  give,  respectively,  straight  horizontal  and  vertical  lines. 

Figs.  61  to  71  give  various  curve  shapes,  corresponding  to 
different  values  of  ft. 


Parabolic  Curves. 

Fig.  01.  n  =  2 

Fig.  62.  ft  =  4 

Fig.  63.  ft  =  8 

Fig.  64.  n  =  \ 

Fig.  65.  n  =  \ 

Fig.  66.  n  =  \ 

Hyperbolic  Curves 


y  =  x2;     the  common  parabola. 
y  =  xi;     the  biquadratic  parabola. 


y  =  x8. 


y=\  x;  again  the  common  parabola. 
y=  l!x;  the  biquadratic  parabola. 
y=</x. 


1 


Fig.  67.     n=  —  1;    y=—',  the  equilateral  hyperbola. 


Fig.  68.     n=-2;    y 


1 


Fig.  69.     n= -A;  y  =  jr 

1  1 

Fig.  70.    ft=-?;  2/=^. 

1  1 

Fig.  71.     n= -j:  U=-^- 


218 


ENGINEERING   MATH  I'M  A  TI(  '8. 


< 
-, 

3 

c 

3 

3 

c 

r 

> 

r 

■J* 

e 

o 

o 

o 

o 

3 

o 

u 

o 

C) 

o 

£1 

Bj 

o 

o3 

i— i 

IN 

ro 

O 

e 

r 

5 

I 

t" 

c 

t 

5 

c 

3 

e 

5 

e 

3 
3 

; 

6 

■ 

3>   9 

o     C 


•-=    .is 


e3 

o    Sh 


-i 

<N      O 
O 

o 


< 

r 

1 

e 

r 

3 

* 
* 

c 

3 

c 

r 

3 

< 

3 

= 

< 

S 

5 

< 

* 

P 

> 
u 

°    o 

"5 


o     — < 
CO 

•J 

CM      ,-' 

d    ^ 


KUl'IHK'AL   CURVES. 


219 


1.4 

1.0 

U.b 

U."D 

0.-4 

/ 

/ 

£.2/ 

1 

0 

2 

0 

4 

0 

6 

0 

,8 

1 

0 

1 

2 

1 

1 

i 

6 

1 

,8 

2 

0 

Fig.  04.     Parabolic  Curve.     y  =  \/.c. 


1,0 

OK 

70^4 

fl-2 

0 

2 

0 

4 

0 

6 

0 

1 

8 

1 

1 

0 

1 

2 

1 

1 

4 

1 

(J 

1 

1 

8 

2 

-  - 

0 

Fig.  6").     Parabolic  Curve.     y=  -yjx. 


220 


BNGINEERINi  /   .1/ .  1 777  EM  A  TJ(  '8. 


l.U 

/0;6 

u;4 

0 

2 

0 

4 

0 

6 

0 

8 

1 

0 

1 

2 

1 

4 

1 

6 

1 

8 

2 

o 

Fig.  66.     Parabolic  Curve.     y="vx. 


•A&- 

24, 

2 -j. 

2-0 

l  i\ 

1-2 

fl-R 

0.-4-" 

0 

4 

0 

8 

1 

2 

i 

1 

e 

2 

0 

2 

4 

2 

8 

3 

2 

3 

0 

4,0 

Fig.  67.     Hyperbolic  Curve  (Equilateral  Hyperbola).    y=    . 


EMPIRICAL  CURVES. 


221 


1-3:2- 

1 

&ib 

■Jfclr 

l.C 

xaS 

\J.o~ 

0=4 

0.4  0.8  1.2  1.6         .2.0  2.4  2.8  3.2  3.6  4.0  4.4 

Fig.  68.     Hyperbolic  Curve.     y=^- 


3£- 

1 

9  R 

4.4 

4iU 

\ 

\ 

U;o 

U;4 

0.4  0.8  1.2  1.6  2.0  2.4  2.8  3.2  3.6  4.0  4.4 


Fig.  69.     Hyperbolic  Curve.     1/  =  ^- 


•  )•)•) 


ENGINEERING   MATHEMATK 'S. 


\ 

\ 

•1 

2.0 

-1.0 

1.6 

0.8 

0.-4 

0 

4 

0 

8 

1 

2 

1 

0 

2 

0 

2 

4 

2 

8 

3 

2 

3 

6 

•4 

0 

4 

t 

Fig.  70.     Hyperbolic  Curve.     y= — — . 


3:2" 

-2r8 

2.1 

\iX 

) 

l.b\ 

\... 

II  s 

0 

4 

0 

8 

1 

•) 

1 

6 

g 

0 

2 

4 

2 

8 

3 

2 

3 

0 

4 

0 

4 

4 

Fig.  71.     Hyperbolic  Curve.    y——=. 

4r 


EMPIRICAL  CURVES.  223 

In  Fig.  72,  sixteen  differcnl  parabolic  and  hyperbolic  curves 
arc  drawn  together  on  the  same  sheet,  for  the  following  values: 
n-1,2,4,8,  oo;  1,1,^,0;     -1,-2,-4,-8;       \,  ~\,    -\. 

147.  Parabolic  and  hyperbolic  curves  may  easily  be  recog- 
nized by  the  fact  that  if  x  is  changed  by  a  constant  factor,  y  also 
changes  by  a  constant  factor. 

Thus,  in  the  curve  y  =  x2,  doubling  the  x  increases  the  y 
fourfold;  in  the  curve  y  =  x159,  doubling  the  x  increases  the  y 
threefold,  etc.;  that  is,  if  in  a  curve, 

y=f(x), 
f(qx) 

-f?~Y =  constant,  for  constant  q,     .     .     .     (17) 

the  curve  is  a  parabolic  or  hyperbolic  curve,  y  =  axn,  and 
f(qx)     a(qx)n 

If  q  is  nearly  1,  that  is,  the  x  is  changed  onjy  by  a  small 
value,  substituting  q  =  l+s,  where  s  is  a  small  quantity,  from 
equation  (18), 

f(x+8X)       „ 

=  (l+s)n  =  l+ns; 


fix) 

hence, 

f(x  +  sr)-f{r) 

fix) 


ns; (19) 


that  is,  changing  x  by  a  small  percentage  s,  y  changes  oy  a  pro- 
portional small  percentage  ns. 

Thus,  parabolic  and  hyperbolic  curves  can  be  recognized  by 
a  small  percentage  change  of  x,  giving  a  proportional  small 
percentage  change  of  y,  and  the  proportionality  factor  is  the 
exponent  n;  or,  they  can  be  recognized  by  doubling  x  and 
seeing  whether  y  hereby  changes  by  a  constant  factor. 

As  illustration  are  shown  in  Fig.  73  the  parabolic  curves, 
which,  for  a  doubling  of  x,  increase  y:  2,  3,  4,  5,  6,  and  8  fold. 

Unfortunately,  this  convenient  way  of  recognizing  parabolic 
and  hyperbolic  curves  applies  only  if  the  curve  passes  through 
the  origin,  that  is,  has  no  constant  term.  If  constant  terms 
exist,  as  in  equation  (16),  not  x  and  y,  but  (x—c)  and  (y—b) 
follow  the  law  of  proportionate  increases,  and  the  recognition 


224 


ENGINEERING  MA TIIEMATICS. 


becomes  more  difficult;    that  is,  various  values  of  c  and  of  b 
are  to  be  tried  to  find  one  which  gives  the  proportionality. 


0.2  o.i  o.<;  0.8  1.0  1.2  1.1 

Fig.  72.     Parabolic  and  Hyperbolic  Curves.    y=x». 

148.  Taking  the  logarithm  of  equation  (15)  gives 

log  y     log  a  !  //  log  .r;      .     ...     .     .     (20) 


EMPIRICAL  CURVES. 


225 


that  is,  a  si  raight  line :  hence,  a  parabolic  or  hyperbolic  curve  can 
be  recognized  by  plotting  the  logarithm  of  y  against  the  loga- 
rithm of  x.  If  this  gives  a  straight  line,  the  curve  is  parabolic 
or  hyperbolic,  and  the  slope  of  the  logarithmic  curve,  tan  0  =  n, 
is  the  exponent. 


S7 

1 1 

/ 

&o- 

03  / 
III 

<?s  1 

'      *• 

1  ft 

i 

I 

\ 

£/ 

/' 

W/ 

u  / 

liU 

vrd 

0.2  0.4  0.6  0.8  1.0  1.2  1.4 

Fig.  73.     Parabolic  Curves.     y  =  r«. 


1.6 


This  again  applies  only  if  the  curve  contain  no  constant 
term.  If  constant  terms  exist,  the  logarithmic  line  is  curved. 
Therefore,  by  trying  different  constants  c  and  ';,  the  curvature 
of  the  logarithmic  line  changes,  and  by  interpolation  such 
constants  can  be  found,  which  make  the  logarithmic  line  straight, 
and  in  this  way,  the  constants  c  and  b  may  be  evaluated.  If 
only  one  constant  exist,  that  is,  only  b  or  only  c,  the  process  is 
relatively  simple,  but  it  becomes  rather  complicated  with  both 


226  ENGINEER!  VG   MA  THEMATIC 'S. 

constants.     This  fad    makes  it   all   the  more  desirable  to  get 
from  the  physical  nature  of   the  problem   some  idea  on  the 
existence  and  the  value  of  the  constant  terms. 
Differentiating  equation  1 2(  1 1  gives : 

(111       dx 
■      n      ; 

.'/ 

that  is,   in   a  parabolic    or  hyperbolic   curve,   the  percentual 
change,  or  variation  of  y}  is  n  times  the  percentual  change, 
or  variation  of  x,  if  n  is  the  exponent. 
Herefrom  follows : 

dy 

y 


n- 


dx 

x 


that  is,  in  a  parabolic  or  hyperbolic  curve,  the  ratio  ofvariatio  . 
dy 

y 

ra  =  — ,  is  a  constant,  and  equals  the  exponent  n. 

x 

Or,  inversely: 

If  in  an  empirical  curve  the  ratio  of  variation  is  constant 
the  curve  is— within  the  range,  in  which  the  ratio  of  variation 
is  constant — a  parabolic  or  hyperbolic  curve,  which  has  as 
exponent  the  ratio  of  variation. 

In  the  range,  however,  in  which  the  ratio  of  variation  is 
not  constant,  it  is  not  the  exponent,  and  while  the  empirical 
curve  might  be  expressed  as  a  parabolic  or  hyprebolic  curve 
with  changing  exponent  (or  changing  coefficient),  in  this  case 
the  exponent  may  be  very  different  from  the  ratio  of  varia- 
tion, and  the  change  of  exponent  frequently  is  very  much 
smaller  than  the  change  of  the  ratio  of  variation. 

This  ratio  of  variation  and  exponent  of  the  parabolic  or 
hyperbolic  approximation  of  an  empirical  curve  must  not  be 
mistaken  for  each  other,  as  has  occasionally  been  done  in 
reducing  hysteresis  curves,  or  radiation  curves.    They  coincide 


EMPIRICAL  CURVES.  227 

only  in  that  range,  in  which  exponent  n  and  coefficient  a  of 
the  equation  y  =  axn  are  perfectly  constant.  If  this  is  not 
the  case,  then  equation  (20)  differentiated  gives: 

dy    da    .         ,      n 

—  = h  log  x  an  +  -ax, 

y       a  x 

and  the  ratio  of  variation  thus  is: 

dy 

1 1  x  da  dn 

m  =  -f-  =n+-      -  +  x  log  x  — ; 
(lx  ax  x 

x 
that  is,  the  ratio  of  variation  m  differs  from  the  exponent  n. 

Exponential  and  Logarithmic  Curves. 

149.  A  function,  which  is  very  frequently  met  in  electrical 
engineering,  and  in  engineering  and  physics  in  general,  is  the 
exponential  function, 

y  =  aenx\ (21) 

which  may  be  written  in  the  more  general  form, 

y-b  =  as"u-c) (22) 

Usually,  it  appears  with  negative  exponent,  that  is,  in  the 
form, 

y  =  ae~nx (23) 

Fig.  74  shows  the  curve  given  by  the  exponential  function 
(23)  for  a  =  l;  n=l\  that  is, 

*/=£"*, (24) 

as  seen,  with  increasing  positive  x,  y  decreases  to  0  at  x=  +  00, 
ancTwith  increasing  negative  x,  y  increases  to  a>  at  x=  —  00. 


22S 


ENGINEERING  MATHEMATICS. 


The  curve,  ?y=£+r,  has  the  same  shape,  except  that  the 
positive  and  the  negative  side  (right  and  left)  are  interchanged. 

Inverted  these  equations  (21)  to  (24)  may  also  be  written 
thus, 

nx  =  log— : 
°  a' 

?i(x-c)=  log— — ; 


nx  ■■ 


a 


log  — 

^a' 


i  = -logy; 
that  is,  as  logarithmic  curves. 


•a.O        -1.6        -1.2        -0.8        -0,4  0  0.4  0.8  1.2  1.6 

Fig.  74.     Exponential  Function,     y^e-*. 


(25) 


m- 

-t* 

c) 

1.2 

"  n 

1.0 

4-0 

0;8 

■4-n 

H);6 

?  0 

0.4 

1-0 

u.~ 

2.0 


150.  The  characteristic  of  the  exponential  function  (21)  is, 
that  an  increase  of  x  by  a  constant  term  increases  (or,  in  (23) 
and  (24),  decreases)  y  by  a  constant  factor. 

Thus,  if  an  empirical  curve,  y=f{x),  has  such  characteristic 

that 


/'•'•  \-q) 

•  /./  >.    =  constant,  for  constant  q, 


(26) 


EMPIRICAL  CURVES. 


229 


the  curve  is  an  exponential  function,  y  =  aenx,  and  the  following 
equation  may  be  written : 


f(x  +  q)      q£"(*+g> 
f(x)  aenx~ 


_  ffng 


(27) 


Hereby  the  exponential  function  can  easily  be  recognized, 
and  distinguished  from  the  parabolic  curve;  in  the  former  a 
constant  term,  in  the  latter  a  constant  factor  of  x  causes  a 
change  of  y  by  a  constant  factor. 

As  result  hereof,  the  exponential  curve  with  negative 
exponent  vanishes,  that  is,  becomes  negligibly  small,  with  far 
greater  rapidity  than  the  hyperbolic  curve,  and  the  exponential 


pl-Oj 

-0.8 

v£_i 

£ 

2.4 

0:6 

(  x+ 

5)*S 

0;4- 

2.4 

(  JC+1  r>" 

2 

£  ~ 

aT^ 

0:2 

0.4  0.8  1.2  1.6  2.0  2.4  2.8  3.2  3.6 

Fig.  75.     Hyperbolic  and  Exponential  Curved  Comparison. 


4.0 


function   with    positive   exponent   reaches   practically    infinite 
values   far   more   rapidly  than   the   parabolic   curve.     This   is 


75, 


in    which    are    shown    superimposed 
~x,    and    the    hyperbolic    curve, 


illustrated    in    Fig. 

the  exponential    curve,    y- 

2.4  ... 

y  =  - ,  „,  ,.    which    coincides    with   the    exponential    curve 

(z  +  1.55)2  x 

at  j  =  0  and  at  x  =  l. 

Taking  the  logarithm  of  equation  (21)  gives  hgy=^hga  + 

nx  log  e,  that  is,  log  y  is  a  linear  function  of  x,  and  plotting 

log  y  against  x  gives  a  straight  line.     This  is  characteristic  of 


230 


A'.\  ( SNEERING   MATH  EM  A  TK  <S. 


the  exponential  functions,  and  a  convenient  method  of  recog- 
nizing them. 

However,  both  of  these  characteristics  apply  only  if  x  and  y 
contain  no  constant  terms.  With  a  single  exponential  function, 
oiilv  the  constant  term  of  y  needs  consideration,  as  the  constant 
term  of  x  may  be  eliminated.  Equation  (22)  may  be  written 
thus : 

y—b=aeMx-c') 


,ae-ncenx 


=At 


nx 


(28) 


where  A=ae~nc  is  a  constant. 

An  exponential  function  which  contains  a  constant  term  b 
would  not  give  a  straight  line  when  plotting  log  y  against  x, 


\ 

-hi 

\ 

\ 

(i)  y=e-x+o,5s-2x 

(2)  y=£-^+o.2e~2x 
(3)y=e-3D 

(4)  y=  £-£C_o.2£-2a; 

(5)  2/=£"£C-0.5£"2a; 

(6)  y=E-OC-0,8£-2X 

(7)  y  =  e-X-£-*x 

(8)  Z/=£-a5-i.f£-2a3 

-1t2 

\  \ 

(1) 

Vj)\ 

-1.0 

\s\  \ 

\ 

-0,8 

\(4 

\  ^ 

-0-.fr 

t\ 

0.-4 

(6), 

-0.2 

/ 

<7> 

/ 

/ 

9- 

i 

0 

8 

1 

2 

1 

(j 

2 

0 

2 

1 

2 

8 

0 

1 

/(h) 

-0.2 

-0.4 

1 

Fig.  7(3.     Exponential  Functions. 


EM  PI  Ml  AL  CURVES. 


231 


but  would  give  a  curve.  In  this  case  then  log  (y— b)  would  be 
plotted  against  x  for  various  values  of  b,  and  by  interpolation 
that  value  of  b  found  which  makes  the  logarithmic!  curve  a 
straight  line. 

151.  While  the  exponential  function,  when  appearing  singly, 
is  easily   recognized,   this   becomes   more   difficult   with   com- 


(l)  y=  £~x+o.5e~wx 
(•>)  y=e  x 

(3)  y=  £~X-0.l£~iOX 

(4)  Z/=£~a?-n.5£"lox 

(5)  ?/=£-«- £-1003 

(«)  y=  e~x-i.5£~mx 

1(1 

) 

v\ 

CfNj 

T 

\ 

h 

/-^ 

t 

) 

\ 

1 

0 

4 

0 

.8 

1 

2 

1 

6 

2 

0 

2 

4 

9 

,8 

1.4 


1.2 


1.0 


0.8 


0.G 


0.4 


-0.2 


■0.4 


Fig.  77.     Exponential  Functions. 

binations  of  two  exponential  functions  of  different  coefficients 
in  the  exponent,  thus, 

y=a1e-cix±a2£~ctx,       (29) 

since  for  the  various  values  of  a\,  a2,  c\,  c>,  quite  a  number  of 
various  forms  of  the  function  appear. 

As  such  a  combination  of  two  exponential  functions  fre- 
<  •  -entry  appears  in  engineering,  some  of  the  characteristic  forms 
are  plotted  in  Figs.  70  to  78. 


232  ENGINEERING  MATHEMATICS. 

Fig.  76  gives  the  following  combinations  of  $~x  and  s~2x: 

(1)  y=e-z+0.5s-2*; 

(2)  y=e-*+0.2.-*»; 

(3)  y=e"*; 

(4)  y=e-*-0.2e-a.; 

(5)  7/=c---r-0.5c-2^; 
(G)     y^s-^-O.Sc--2^; 

(7)       2/=£-x_£-2x; 

(8)     ^=  e-*-  1.5s-23-. 


AsUS 

cosha;  =  jie+ar-£~a;} 
sinho^ne+^-e-2'} 

/ 

/ 

/ 

/ 

oy 

W 

^s* 

.C 

1.0 

-/ 

IV'! 

• 

~U:4 

0;Z 

0 

^ 

0 

1 

0 

f> 

0 

s 

1 

0 

1 

g 

i 

4 

Fig.  78.     Hyperbolic  Functions. 


EMPIRICAL  CURVES.  233 

Fig.  77  gives  the  following  combination  of  s~x  and  s_10i ; 

(1)  2/=e-*+0.5s-10*; 

(2)  y-.1*; 

(3)  ?/=c---0.l£-10^; 

(4)  ?/=c--0.or10-; 

(5)  y=.e-*_ff-iOr. 


(6)     y-e-*-l. 


}£ 


-10. 


Fig.  78  gives  the  hyperbolic  functions  as  combinations  cf 
e+x  and  e~x  thus, 

2/  =  cosh  .r  =  K^  +  J"  +  ^_:r); 

?/  =  sinh  x  =  i(c  +  -r—  £-j). 


C.  Evaluation  of  Empirical  Curves. 

152.  In  attempting  to  solve  the  problem  of  finding  a  mathe- 
matical equation,  y=f(x),  for  a  series  of  observations  or  tests, 
the  corresponding  values  of  x  and  y  are  first  tabulated  and 
plotted  as  a  curve. 

From  the  nature  of  the  physical  problem,  which  is  repre- 
sented by  the  numerical  values,  there  are  derived  as  many 
data  as  possible  concerning  the  nature  of  the  curve  and  of  the 
function  which  represents  it,  especially  at  the  zero  values  and 
the  values  at  infinity.  Frequently  hereby  the  existence  or 
absence  of  constant  terms  in  the  equation  is  indicated. 

The  log  x  and  log  y  are  tabulated  and  curves  plotted  between 
x,  y,  log  x,  log  y,  and  seen,  whether  some  of  these  curves  is  a 
straight  line  and  thereby  indicates  the  exponential  function,  or 
the  parabolic  or  hyperbolic  function. 

If  cross-section  paper  is  available,  having  both  coordinates 
divided  in  logarithmic  scale,  and  also  cross-section  paper  having 
one  coordinate  divided  in  logarithmic,  the  other  in  common 
scale,  x  and  y  can  be  directly  plotted  on  these  two  forms  of 
ogarithmic  cross-s  ction  paper.  Usually  not  much  is  saved 
thereby,  as  for  the  n  merical  calculation  of  the  constants  the 
logarithms  still  have  to  be  tabulated. 


23  1  ENGINEERING  MATHEMATH  'S. 

If  neither  of  the  four  curves:  x,  y;  x,  log  y;  log  x,  y;  log  x, 
logy  is  a  straight  line,  and  from  the  physical  condition  the 
absence  of  a  constant  term  is  assured,  the  function  is  neither 
an  exponential  nor  a  parabolic  or  hyperbolic.  If  a  constant 
term  is  probable  or  possible,  curves  are  plotted  between  .r, 
y—b,  log  j,  log  (y—b)  for  various  values  of  b,  and  if  hereby 
one  of  the  curves  straightens  out,  then,  by  interpolation, 
that  value  of  b  is  found,  which  makes  one  of  the  curves  a  straight 
line,  and  thereby  gives  the  curve  law.  In  the  same  manner, 
if  a  constant  term  is  suspected  in  the  x,  the  value  (x-c)  i< 
used  and  curves  plotted  for  various  values  of  c.  Frequently  the 
existence  and  the  character  of  a  constant  term  is  indicated  by 
the  shape  of  the  curve;  for  instance,  if  one  of  the  curves  plotted 
between  .r,  y,  log  x,  log  y  approaches  straightness  for  high,  or  for 
low  values  of  the  abscissas,  but  curves  considerably  at  the 
other  end,  a  constant  term  may  be  suspected,  which  becomes 
less  appreciable  at  one  end  of  the  range.  For  instance,  the 
effect  of  the  constant  c  in  (x-c)  decreases  with  increase  of  x. 

Sometimes  one  of  the  curves  may  be  a  straight  line  at  one 
end,  but  curve  at  the  other  end.  This  may  indicate  the  presence 
of  a  term,  which  vanishes  for  a  part  of  the  observations.  In 
this  case  only  the  observations  of  the  range  which  gives  a 
straight  line  are  used  for  deriving  the  curve  law,  the  curve 
calculated  therefrom,  and  then  the  difference  between  the 
calculated  curve  and  the  observations  further  investigated. 

Such  a  deviation  of  the  curve  from  a  straight  line  may  also 
indicate  a  change  of  the  curve  law,  by  the  appearance  of 
secondary  phenomena,  as  magnetic  saturation,  and  in  this  case, 
an  equation  may  exist  only  for  that  part  of  the  curve  where  the 
secondary  phenomena  are  not  yet  appreciable.  The  same 
equation  may  then  be  applied  to  the  remaining  part  of  the  curve, 
by  assuming  one  of  the  constants,  as  a  coefficient,  or  an  exponent, 
to  change.  Or  a  second  equation  may  be  derived  for  this  pail 
of  the  curve  and  one  part  o  the  curve  represented  by  one,  the 
other  by  another  equat  on.  The  two  equations  may  then  over- 
lap, and  at  some  point  the  curve  represented  equally  well  by 
either  equation,  or  the  ranges  of  application  of  the  two  equa- 
tions may  be  separated  by  a  transition  range,  in  vhich  neither 
applies  exactly. 


EMPIRICAL  CURVES.  235 

If  neither  the  exponential  "functions  nor  the  parabolic  and 
hyperbolic    curves    satisfactorily    represent    the    observations, 

x 
further  trials  may  be  made  by  calculating  and  tabulating  • 

5  y 

1     V  ,  ,  ,  It/.. 

and  -,  and  plot  tine;  curves  between  x,  y,  —,  — .     Also  exoressions 

x  °  y   x 

as  x2  +  by2,  and  (x  —  a)2+h(y—c)2,  etc.,  may  be  studied. 

Theoretical  reasoning  based  on  the  nature  of  the  phenomenon 
represented  by  the  numerical  data  frequently  gives  an  indi- 
cation of  the  form  of  the  equation,  which  is  to  be  expected, 
and  inversely,  after  a  mathematical  equation  has  been  derived 
a  trial  may  be  made  to  relate  the  equation  to  known  laws  and 
thereby  reduce  it  to  a  rational  equation. 

In  general,  the  resolution  of  empirical  data  into  a  mathe- 
matical expression  largely  depends  on  trial,  directed  by  judg- 
ment based  on  the  shape  of  the  curve  and  on  a  knowledge  of 
the  curve  shapes  of  various  functions,  and  only  general  rules 
can  thus  be  given. 

A  number  of  examples  may  illustrate  the  general  methods  of 
reduction  of  empirical  data  into  mathematical  functions. 

153.  Example  1.  In  a  118-volt  tungsten  filament  incan- 
descent lamp,  corresponding  values  of  the  terminal  voltage  e 
and  the  current  i  are  observed,  that  is,  the  so-called  "  volt- 
ampere  characteristic  "  is  taken,  and  therefrom  an  equation  for 
the  volt-ampere  characteristic  is  to  be  found. 

The  corresponding  values  of  e  and  i  are  tabulated  in  the 
first  two  columns  of  Table  III  and  plotted  as  curve  I  in 
Fig.  79.  In  the  third  and  fourth  column  of  Table  III 
are  given  log  e  and  log  i.  In  Fig.  79  then  are  plotted 
log  e,  i,  as  curve  II;  e,  log?,  as  curve  III;  log  e,  log?',  as 
curve   IV. 

As  seen  from  Fig.  79,  curve  IV  is  a  straight  line,  that  is 

log  i  --=  A  +  n  log  e ;     or     i  =  aen, 

which  is  a  parabolic  curve. 

The  constants  a  and  n  may  now  be  calculated  from 
the    numerical  data    of    Table    III    by  the    method    of    least 


236 


ENGINEERING  M  A  Til  EM  A  TICS. 


squares,  as  discussed  in   Chapter  IV,    paragraph  120.      While 
this  method  gives  the  most  accurate  results,  it  is  so  laborious 


0.2      0.4      0.C      0.8      1.0      1.2      1.4      l.G      1.8      2.0      2.2      2A=Jog  e 


2 

J         4 

J         I 

0 

80       100      1 

f° 

i 

0      100     180      200     2 

ta=e 

,       Zogf  i 

ji 

■*■       j^T 

9.0 

~~**®^~~ 

fSJ 

III 

iSJ 

9.5 

& 

r 

*  \  J~ 

r 



9.4 

9.3 

^IV 

/ 

® 

9.2 

P 

sr 

9.1 

i        9.0 

f 

\j 

0.45     8.9 

j7 

\Or 

0.40     1.8 

4 

XV 

\^> 

0.35     8.7 

0.30     3.0 

n\ 

0.25  8.5 
0.20     8.4 

i 

$ 

e*^> 

0.1")  8.3 
0.10  8.2 
0.05     8.1 

/4 

r 



FlG.   79.     Investigation   of   Volt-ampere   Characteristic   of  Tungsten    Lamp 

Filament. 


as  to  be  seldom  used  in  engineering;  generally,  values  of 
the  constants  a  and  n,  sufficiently  accurate  for  most  practical 
purposes,  arc  derived   by  the  following   method: 


EMPIRICAL    CURVES. 


237 


Table  III. 
VOLT-AMPERE  CHARACTERISTIC  OF  118-VOLT  TUNGSTEN  LAMP. 


e 

i 

log  e 

logi 

8-211+0-6  log  e. 

J 

2 

0   0245 

0   301 

8-392 

8-389 

-0003 

4 

0   037 

0   602 

8  568 

8-572 

-0004 

8 

0  0568 

0-903 

8-754 

8   753 

+  0-001 

16 

0-0855 

1-204 

8-932 

8   933 

-0001 

25 

01125 

1-398 

9051 

9  050 

+  0001 

32 

0-1295 

1-505 

9112 

9114 

-0002 

50 
64 

01715 
0-200 

1.699 

9-234 
9.301 

9-230 
9-295 

+  0-004 
+  0006 

1.806 

100 

0-2605 

2  000 

9-416 

9-411 

+  0  005 

125 

02965 

2097 

9-472 

9-469 

+  0003 

150 

0   3295 

2176 

9-518 

9518 

0 

180 

0-3635 

2-255 

9-561 

9- 564 

-0003 

200 

0   3865 

2-301 

9-587 

9-592 

-0  005 

218 

0-407 

2-338 

9-610 

9-614 

-0-004 

2"7=    7-612 

2043 

avg.  ±0003 

i 

'=  14973 

6-465 

4.  7  per  cent 

=     7361 

4-422 

4-422 

n=    = 

7-361 

0.6007  =  0 

6 

i"14=  22-585 

8-505 

06X22585       = 

=       13   551 

•551  =  4.954 

i  =  8. 505-13 

4  954  +  14  = 

8-211 

logi'=§. 211+0-6  1 

ag  e     and     i  = 

0-01625''0'6 

The  fourteen  sets  of  observations  are  divided  into  two 
groups  of  seven  each,  and  the  sums  of  log  e  and  log?'  formed. 
They  are  indicated  as  H7  in  Table  III. 

Then  subtracting  the  two  groups  E7  from  each  other, 
eliminates  A,  and  dividing  the  two  differences  J,  gives  the 
exponent,  n=0.6011;  this  is  so  near  to  0.6  that  it  is  reasonable 
to  assume  that  /z  =0.6,  and  this  value  then  is  used. 

Now  the  sum  of  all  the  values  of  log  e  is  formed,  given  as 
£14  in  Table  III,  and  multiplied  with  n=0.6,  and  the  product 


•_>:  ;s  ENGINEERING  M.  1  THEM  A  TI<  'S. 

subtracted   from  the  sum  of  all  the  logi.    The  difference  J 
then  equals  1  1-1,  and,  divided  by  11,  gives 

A  =  log  a  =  8.211; 

hence,  a=0.01625,  and  the  volt-ampere  characteristic  of  this 
tungsten  lamp  thus  follows  the  equation, 

log  i=8.211+0.6  log  e; 

7;  =  0.0K)25e°-G. 

From  e  and  i  can  be  derived  the  power  input  p=--ei,  and  the 

e 

resistance  r  =  — : 
^ 

p  =  0.01025^  -6; 


0.01625' 

and,  eliminating  e  from  these  two  expressions,  gives 

2>=0.01()255r4  =  11.35r4Xl0-10, 

that  is,  the  power  input  varies  with  the  fourth  power  of  the 
resistance. 

Assuming  the  resistance  r  as  proportional  to  the  absolute 
temperature  T,  and  considering  that  the  power  input  into  the 
lamp  is  radiated  from  it,  that  is,  is  the  power  of  radiation  Pr, 
the  equation  between  p  and  r  also  is  the  equation  between  Pr 
and  T,  thus, 

thai  is,  the  radiation  is  proportional  to  the  fourth  power  of  the 
absolute  temperature.  This  is  the  law  of  black  body  radiation, 
and  above  equation  of  the  volt-ampere  characteristic  of  the 
tungsten  lamp  thus  appears  as  a  conclusion  from  the  radiation 
law.  that  is,  as  a  rational  equation. 

154.  Example  2.  In  a  magnetite  arc,  at  constant  arc  length, 
the  voltage  consumed  by  the  arc,  e,  is  observed  for  different 
values  of  current  i.  To  find  the  equation  of  the  volt-ampere 
characteristic  of  the  magnet'te  arc: 


EMPIRICAL  CURVES. 


239 


Table  IV. 
VOLT-AMPERE  CHARACTERISTIC  OF  MAGNETITE  ARC. 


t 

e 

log  )' 

log 

(e-401 

log  (p-40 

(e-30) 

log 30" 

ec 

J 

0-5 

160 

9- 699 

2-204 

120 

2- 079 

130 

2114 

158 

-2 

1 

120 

0  000 

2-079 

80 

1-903 

90 

1   954 

120-4 

+  04 

2 

94 

0301 

1-973 

54 

1-732 

64 

1   806 

94 

0 

4 

75 

0602 

1-875 

35 

1-544 

45 

1    653 

75-2 

+  0-2 

8 

62 

0-903 

1792 

22 

1-342 

32 

1    505 

62 

0 

12 

56 

1   079 

1-748 

1G 

1-204 

26 

1-415 

56- 2 

+  0-2 

^3  =  0-000 5   874 

^3  =  2-584 4- 573 

J  =  2  •  584 -1-301 

—  1   301 

71=  -  =  -0-5034-  -0-5 

2-584 

^6  =  2-584 10-447 

2 •  584 X -0-5 =  -1-292 

d=    11-739 
11   739  ^6=      1-956=1 

log  (c-30)  =  l-C56-0-5logi 

90  4 
e-30   =90-4i-  »'=     and      p  =  30+      1_ 

N      ! 


The  first  four  columns  of  Table  IV  give  i,  e,  log?',  log  e. 
Fig.  SO  gives  the  curves:  i,  e,  as  I;  i,  log  e,  as  II;  hgi,  e,  as 
III:  log  i,  log  e,  as  IV. 

Neither  of  these  curves  is  a  straight  line.  Curve  IV  is 
relatively  the  straightest,  especially  for  high  values  of  e.  This 
points  toward  the  existence  of  a  constant  term.  The  existence 
of  a  constant  term  in  the  arc  voltage,  the  so-called  "  counter 
e.m.f.  of  the  arc  "  is  physically  probable.  In  Table  IV  thus 
are  given  the  values  (e-40)  and  log  (e- 40),  and  plotted  as 
curve  V.  This  shows  the  opposite  curvature  of  IV.  Thus  the 
constant  term  is  less  than  40.  Estimating  by  interpolation,  and 
calculating  in  Table  IV  (e-30)  and  log  (e-30),  the  latter, 
plotted  against  log?'  gives  the  straight  line  VI.     The  curve  law 

thus  is 

log  (e-30)  =  A+n  log  i. 


240 


ENGINEERING    MA  Til  EM  A  TI<  'S. 


Proceeding  in  Table  IV  in  the  same,  manner  with  logi 
and  log  (e— 30)  as  was  done  in  Table  III  with  log  e  and  log  i, 
gives 

n=-0.5;     .4  =  log  a  =  1.956;    and     a=90.4; 


loge 


Fig.  80.     Investigation  of  Volt-ampere  Characteristic  of  Magnetite  Arc. 

hence 

log  (e-  30)  =  1 .056  -  0.5  log  i; 

e-30  =  90.4i-°'5; 


EMPIRICAL  CURVES. 


241 


which  is  the  equation  of  the  magnetite  arc  volt -ampere  charac- 
teristic. 

155.  Example  3.  The  change  of  current  resulting  from  a 
change  of  the  conditions  of  an  electric  circuit  containing  resist- 
ance, inductance,  and  capacity  is  recorded  by  oscillograph  and 
gives  the  curve  reproduced  as  I  in  Fig.  81.     From  this  curve 


log 

*^v 

\v 

3 C 

>-  ^ 

<V 

~N 

i 

A 

\ 

) 

\ 

> 

u 

iiU 

\< 

*r 

II 

\<r 

l.u 

X 

\y 

IIl\ 

\> 

y.o 

\ 

\ 

l\ 

u.o" 

>^ 

0:4 

0 

4 

0 

8 

1 

2 

t 
1 

6 

2 

0 

9 

4 

2. 

8 

Fig.  81.     Investigation  of  Curve  of  Current  Change  in  Electric  Circuit. 

are  taken  the  numerical  values  tabulated  as  t  and  i  in  the  first 
two  columns  of  Table  V.  In  the  third  and  fourth  columns  are 
given  log/  and  log/,  and  curves  then  plotted  in  the  usual 
manner.  Of  these  curves  only  the  one  between  t  and  log  i 
is  shown,  as  II  in  Fig.  81,  since  it  gives  a  straight  line  for  the 
higher  values  of  /.     For  the  higher  values  of  /,  therefore, 

log  i  =  A  —  nt ;    or,     i =ae 
that  is,  it  is  an  exponential  function. 


-nt. 


242 


ENGINEERING    U  AT  II  EM  AT  H  'S. 


Table  V. 

TRANSIENT  CURRENT  CHARACTERISTICS. 


t 

i 

log/ 

log  i 

ii 

i' 

t            log  i'          fa 

ic 

J 

0 

2.10 

0-322 

4.94 

2.84 

0           0.461     2.85 

2.09 

-0.01 

CI 
0.2 

2-48 
2. 66 

9.000 
9.301 

0.394 
C425 

4.44 
3.98 

1  .96 
1  .32 

0-1       0-29.      1.94 

2.50 
2-66 

+  0.02 
0 

0-2       0-121      1.32 

0.4 
0.8 

2.58 
2.00 
1.36 

§■602 
9.903 

0.079 

C412 
0.301 

3.21 
2.09 
1.36 

0.63 
0  .09 
0 

0-4       9-799      0.61 

2.60 
1  .96 
1.33 

+  0.02 
-0.04 
-0.03 

0-8       8-954     0.13 
—        0.03 

L2 

0.134 

1  .6 

0.90 

0.204 

9.954 

0.89 

-o.oi 

0.01 

0.88 

-0.02 

2.0 
2.5 

0.58 
0.34 

0  .301 
0.398 

9.763 

0.58 
0.34 

0 
0 

0.58 
0.34 

0 
0 

9-531 

3.0 

0-20 

0.477 

9-301 

0.20 

0 

0.20 

0 

23  = 

4   8 

9.851 

J2 

-0.1     0.753 

V    

*"  2  — 

4  8 
—=1.6 

3 

5.5 

9-851     _ 

-  =  9.95C 
3 

9.832 

2i 

J 

-0.6     9.920 

=  0.5-0.833 

5.5_ 
2 

J  = 

log  rXl 

2.75 

9   832     _ 

-=9.41( 
2 

-0.53' 
0.499 

i         log 

t 

n 

eX0.5  =  0.217 

-0.833 
8=-  =  -3.84 

0.217 

=  1.15 
.15  = 

7M  = 

0.534 

-=  -1  .07 
0.499 

y 

,    =    0.7     0.673 

25= 

10.3 

8-683 

II!    lug 

£X0.7  = 
J  = 

=  -1.167 

=       1.840 

10  .3  X/ti  log  i 

J  = 

-4.78 

4 

!7 

1  .840  -:-  4  =  0  .460  =  -U>-  log  as 
n?=2.85 

3   4( 

3.467^5  =  ( 

I  693  =  ^!  =  ^  (ii 

log  w=0  .460-3  .84'  log  £ 

<v,  =  4 

.94 

'-'  =  2  .85£—  »■**' 

log  ii=( 

1.693-1  .07'  log  £ 

ii  =  4 

i.94£-i-07< 

»c  =  4. 94c " 

-  1.07  _ 

2.85s-3"84' 

To  calculate  the  constants  a  and  n,  the  range  of  values  is 
used,  in  which  the  curve  II  is  straight;  that  is,  from  £  =  1.2 
to  /  =  3.  As  these  arc  five  observations,  they  are  grouped  in  two 
pairs,  the  first  3,  and  the  last  2,  and  then  for  /  and  log?',  one- 
third  of  the  sum  of  the  first  3,  and  one-half  of  the  sum  of  the 
last  2  are  taken.     Subtracting,  this  gives, 

J/-  1.15;     A  log %=  -0.534. 

Since,  however,  the  equation,  i=ae"  "',  when  logarithmated, 
gives 

log  i=log  a  -  ill  log  e, 
thus  J  log  i=—n  log  eJt3 


EMPIRICAL  CURVES.  243 

it  is  necessary  to  multiply  J/  by  log  e  =  0.4343  before  dividing  it 
into  log  i  to  derive  the  value  of  n.     This  gives  ft =1.07. 

Taking  now  the  sum  of  all  the  five  values  of  £,  multiplying  it 
by  log  e,  and  subtracting  this  from  the  sum  of  all  the  five  values 
of  log  i,  gives  5A  =  3.4(37;   hence 

A  =  log  a  =  0.693, 

a  =  4.94, 

and  log  ix  =0.693  -1.07/  log  e; 

n  =  4.94c--107'. 

The  current  i\  is  calculated  and  given  in  the  fifth  column 
of  Table  V,  and  the  difference  i'  =  A  =  ii—i  in  the  sixtli 
column.  As  seen,  from  £  =  1.2  upward,  %i  agrees  with  the 
observations.  Below  £=1.2,  however,  a  difference  %'  remains, 
and  becomes  considerable  for  low  values  of  /.  This  difference 
apparently  is  due  to  a  second  term,  which  vanishes  for  higher 
values  of  t.  Thus,  the  same  method  is  now  applied,  to  the 
term  i';  column  8  gives  log/,  and  in  curve  III  of  Fig.  SI  is 
plotted  log  i'  against  t.  This  curve  is  seen  to  be  a  straight 
line,  that  is,  %'  is  an  exponential  function  of  t. 

Resolving  i'  in  the  same  manner,  by  using  the  first  four 
points  of  the  curve,  from  £  =  0  to  £  =  0.4,  gives 

log i2 = 0.460 -3.84£ log  e; 

i2  =  2.85  c- ~3-84'' 
and,  therefore, 

i  =  i1-V2  =  4.94c--107'-2.85c--3-84i 

is  the  equation  representing  the  current  change. 

The  numerical  values  are  calculated  from  this  equation 
and  given  under  ic  in  Table  V,  the  amount  of  their  difference 
from  the  observed  values  are  given  in  the  last  column  of  this 
table. 

A  still  greater  approximation  may  be  secured  by  adding 
the  calculated  values  of  io  to  the  observed  values  of  i  in  the 
last  five  observations,  and  from  the  result  derive  a  second 
approximation  of  i\,  and  by  means  of  this  a  second  approxi- 
mation of  i-2. 


244 


ENGINEERING   MA  Til  EM  A  TICS. 


156.  As  further  example  may  be  considered  the  resolution 
of  the  core  loss  curve  of  an  electric  motor,  which  had  been 
expressed  irrationally  by  a  potential  series  in  paragraph  144 
and  Table  I. 

Table  VI. 
CORE  LOSS   CURVE. 


e 
Volts. 

Pi  kw. 

log  e 

log  Pi 
9799 

1-6  log  e 

.1  =  log  Pi 

—  1.6  log  e 

Pc 

J 

40 

063 

1-602 

2563 

7-236 

0-70 

"0-07 

60 

1 

36 

1778 

0   134 

2 

845 

7   289  1 
7-293        avg. 

1 

34 

+  0 

02 

80 

2 

18 

1.903 

0  338 

3 

045 

2 

12 

+  0 

06 

100 

3 

00 

2.000 

0   477 

3 

200 

7-277  [  7-282 

3 

03 

-0 

03 

120 

3 

93 

2.079 

0.594 

3 

326 

7-268  J 

4 

05 

-0 

12 

140 

6 

22 

2.146 

0.794 

3 

434 

7-360 

5 

20 

+  1 

02 

160 

8 

59 

2-204 

0-934 

3 

526 

7-408 

6 

43 

+  2 

16 

*a 

=  5-283 

0-271 

log  Pj  =  7.  282 +  1.6  log  c 

^3- 

-a 

22 

=  1-761 
=  4079 

0090 
1071 

-Pi=l-914t'''6,  in  watts 

22- 

-2 

A 

=  2  0395 
=  0  2785 

0535 
0-445 

r 

=  0-445 
0-2785 

=  1598~ 

16 

J 

The  first  two  columns  of  Table  VI  give  the  observed  values 
of  the  voltage  e  and  the  core  loss  Pi  in  kilowatts.  The  next 
two  columns  give  log  e  and  log  r\.  Plotting  the  curves  shows 
that  loge,  log  Pi  is  approximately  a  straight  line,  as  seen  in 
Fig.  82,  with  the  exception  of  the  two  highest  points  of  the 
curve. 

Excluding  therefore  the  last  two  points,  the  first  five  obser- 
vations give  a  parabolic  curve. 

The  exponent  of  this  curve  is  found  by  Table  VI  as 
n=  1.598;  that  is,  with  sufficient  approximation,  as  n=1.6. 

To  see  how  far  the  observations  agree  with  the  curve,  as 
given  by  the  equation, 


P*= 


ae 


1.6 


in  the  fifth  column  1.6  log  e  is  recorded,  and  in  the  sixth  column, 
A=loga=log Pi- 1.6 loge.  As  seen,  the  first  and  the  last 
two  values  of  A  differ  from  the  rest.     The  first  value  corre- 


EMPIRICAL  CURVES. 


245 


sponds  to  such  a  low  value  of  Pi  as  to  lower  the  accuracy  of 
the  observation.  Averaging  then  the  four  middle  values, 
gives  ^.  =  7.282;  hence, 

IogP;=7.282  +  1.01oge, 
P;=1.914eL6'  in  watts. 


Fig.  82.     Investigation  of  Cuvres. 

This  equation  is  calculated,  as  Pc,  and  plotted  in  Fig.  82. 
The  observed  values  of  Pi  are  marked  by  circles.  As  seen, 
the  agreement  is  satisfactory,  with  the  exception  of  the  two 
highest  values,  at  which  apparently  an  additional  loss  appears, 
which  does  not  exist  at  lower  voltages.  This  loss  probably  is 
due  to  eddy  currents  caused  by  the  increasing  magnetic  stray 
field  resulting  from  magnetic  saturation. 


24  G 


ENGINEERING  .17.1  THEMATIC 'S. 


157.  As  a  further  example  may  be  considered  the  resolution 
of  the  magnetic  characteristic,  plotted  as  curve  I  in  Fig.  83, 
and  given  in  the  first  two  columns  of  Table  VII  as  JC  and  eft. 

Table  VII. 
MAGNETIC   CHARACTERISTIC. 


JC 

kilolines 

logJC 

log  (B 

(B 

JC 

JC 

"(B 

<BC 

J 

2 

3  0 

0-301 

0-477 

15 

0-667 

6-4 

+  3   4 

4 

8 

4 

0-602 

0 

924 

2 

1 

0-476 

9 

7 

+  13 

e 

11 

2 

0778 

1 

049 

1 

867 

0  536 

11 

6 

+  0-4 

8 

13 

0 

0903 

1 

114 

1 

625 

0  614 

13 

0 

0 

10 

14 

0 

1   000 

1 

146 

1 

40 

0   715 

13 

9 

-0.1 

15 

15 

4 

1   176 

1 

188 

1 

033 

0  974 

15 

45 

+  005 

20 

16 

3 

1  301 

1 

212 

0 

815 

1-227 

16 

3 

0 

30 

17 

2 

1477 

1 

236 

0 

573 

174 

17 

3 

+  01 

40 

17 

8 

1-602 

1 

250 

0 

445 

2- 25 

17 

8 

0 

60 

18 

5 

1-778 

1 

267 

0 

308 

3-25 

18 

4 

-01 

80 

18 

8 

1-903 

1 

274 

0 

235 

4-25 

18-8 

0 

--'4  =  53 

3-530 

^4  =  210 

11-49 

^  =  157 

7  96 

=  00507  =6 

157 

7-96 

-r8  =  263 

263X00507  = 
J  = 

15020 
=13   334 

=    1   686 

1-  686-8  = 

=    0-211  = 

j 

JC 

\     nil     1    f\    r\  r  «■»  ™  T-f           »»»-..!         /I 

i 

JC 

<B 

)  211  +0  0 

\^         iXllX. 

j  — 

0  2 

11  +  0  0507 

JC' 

Plotting  JC,  (B,  log  JC,  log  (B  against  each  other  leads  to  no 
results,  neither  does  the  introduction  of  a  constant  term  do 
this.     Thus  in  the  fifth  and  sixth  columns  of  Table   VII  are 

calculated  —  and  --,  and  are  plotted  against   JC  and  against  CB. 
JC  IB 

3C 

Of  these  four  curves,  only  the  curve  of  —  against    JC  is  shown 

iii  Fig.  83,  as  II.     This  curve  is  a  straight  line  with  the  exception 

of  the  lowest  values;   that  i>, 

JC 


(B 


=  a  +  bX. 


EMPIRICAL  CURVES. 


24; 


Excluding  the  three  lowest  values  of  the  observations,  as 
not  lying  on  the  straight  line,  from  the  remaining  eight  values,, 
as  calculated  in  Table  VII,  the  following  relation  may  be 
derived, 

—  =  0.21 1+0.0507  3C, 

(x> 


l.U 

i.-J 

/n 

O.Q 

SB 

4.0 

/ 

i 

—        /? 

i 

*s.U 

/ 

<" 

/ 

( 

l.j 

/ 

c 

/ 

o 
^-8- 

l.U  / 

a/ 

(jT 

J.GNgp 

1 

D 

i 

i) 

0 

4 

3 

0 

{' 

i) 

1 

1) 

1 

D 

80 

Fig.  S3.     Investigation  of  Magnetization  Curve. 

and  herefrom, 

3C 


(B  = 


0.211+  0.0507  JC 


is  the  equation  of  the  magnetic  characteristic  for  values  of  5C 
from  eight  upward. 

The  values  calculated  from  this  equation  are  given  as  (Bc 
in  Table  VII. 


248  ENGINEERING   MATHEMATICS. 

The  difference  between  the  observed  values  of  — ,  and  the 

(B 

value  given  by  above  equation,  which  is  appreciable  up  to 

5C  =  6,  could  now  be  further  investigated,  and  would  be  found 

to  approximately  follow  an  exponential  law. 

As  a  final  example  may  be  considered  the  investigation  of 
a  hysteresis  curve  of  silicon  steel,  of  which  the  numerical  values 
are  given  in  columns  1  and  2  of  Table  VIII. 

The  first  column  gives  the  magnetic  density  (B,  in  lines  of 
magnetic  force  per  cm.2;  the  second  column  the  hysteresis  loss 
w,  in  ergs  per  cycle  per  kg.  (specific  density  7.5).  The  third 
column  gives  log  (B,  and  the  fourth  column  log  w. 

Of  the  four  curves  between  (B,  w,  log  (B,  log  w,  only  the 
curve  relating  log  w  to  log  (B  approximates  a  straight  line,  and 
is  given  in  the  upper  part  of  Fig.  83a.  This  curve  is  not  a 
straight  line  throughout  its  entire  length,  but  only  two  sections 
of  it  are  straight,  from  (B  =  50  to  (B  =  400,  and  from  (B  =  1600  to 
(B  =  8000,  but  the  curve  bends  between  500  and  1200,  and  above 
8000. 

Thus  two  empirical  formulas,  of  the  form:  w  =  a(S>n,  are 
calculated,  in  the  usual  manner,  in  Table  VIII.  The  one 
applies  for  lower  densities,  the  other  for  medium  densities: 

Low  density :  (B  ^  400 :  w  =  0.0034KB2-11 

Medium  density :  1600  ^  05  ^  8000 :      w  =  O.IOOOCB1-60 

In  Table  VIII  the  values  for  the  lower  range  are  denoted 
by  the  index  1,  for  the  higher  range  by  the  index  2. 

Neither  of  these  empirical  formulas  applies  strictly  to  the 
range:  400  <(B<  1600,  and  to  the  range  (B>8000.  They  may 
be  applied  within  these  ranges,  by  assuming  either  the  coefficient 
a  as  varying,  or  the  exponent  n  as  varying,  that  is,  applying  a 
correction  factor  to  a,  or  to  n. 

Thus,  in  the  range:  400  <(B<  1600,  the  loss  may  be  repre- 
sented by: 

(1)  An  extension  of  the  low  density  formula: 

w  =  a1($>2-11      or      u;=0. 00341  <Bni. 

(2)  An  extension  of  the  medium  density  formula 

u>  =  a2(B1,6        or       w  =  0.1096'  «"*, 


EMPIRICAL   CURVES. 


249 


by  giving  tables  or  curves  of  a  respectively  n.     Such  tables  are 
most  conveniently  given  as  a  percentage  correction. 


L. 

Log 

\\ 

6 

—4 

>^ 

^/ 

LC 

V 

3 

V, 

V 

.' 

2 

o— 

o- 

] 

0 

o- 

1— 

<d:l. 

— , 

^ 

*"- 

/■■ 

1 
— f1 

\ 

s 

/ 

■ 

', 

/ 

5— 

v 

V 

i 

/ 

/N 

\ 

D 

y 

/ 

' 

\ 

_2 

4 

/ 

\ 

\ 

-2 

.A— 

r-^V 

(— f t- 

•— 

«. 

r 

1    / 

•> 

o 

-^ 

"'' 

T 

-1 

lll\ 

-1 

-fc- 

u- 

'.J 

1 

fa 

1 

Lo 

'  P1  = 

2.0 

1 

3 

X) 

4lo 

i 

Fig.  83a. 


The  percentage  correction,  which  is  to  be  applied  to  ax  and 
a2  respectively,  to  nx  and  n2,  to  make  the  formulas  applicable 


250 


ENGINEERING   MA  THEMATK  'S. 


Table  VIII. 
HYSTERESIS  OF  SILICON  STEEL. 


Ja\          Sa'i 

J  m         J  m 

(B 

w 

log® 

log  w 

o\             ai 

in 

n> 

in 

712 

m 

1-544 
1-699 

i 

+  0.46 
-0.03 

■  , 

35 
50 

6.4 
13 

0.806 

1  .117 

+    3  -C 

— 

2.120 

— 

2.03 

-    0.2 

2.109 

60 

19 

1-778 

1.279 

-    1.1 

— 

-0.13 

— 

2.107 

2.14 

80 

36 

1.903 

1  .556 

+    1  .9 

— 

+  0.20 

— 

2.114 

2.12 

100 

57 

2   000 

]  .752 

-    0.2 

— 

-0.02 

— 

2.110 

2.09 

120 
160 

83 
156 

2.079 

1    922 

+    0.7 
+    2.3 

+  0.07 
+  0.21 

2.111 
2.114 

2.16 
2.10 

2.204 

2.193 

200 

245 

2.301 

2   389 

+    0.2 

+  0.02 

— 

2.110 

2. 07 

250 

394 

2.398 

2.595 

+    0.7 

+  0.06 

— 

2.111 

2.03 

300 

571 

2.477 

2.757 

-    0.5 

-0.04 

— 

2.109 

1   98 

400 
500 

1025 
1610 

2.602 

3  .011 

-    2.7 

— 

-0-22 

-3.52 

2.105 

2.03 
2.02 

2.699 

3.207 

-    4.7 

-29.5 

-0.37 

2.102 

1.544 

600 

2320 

2.778 

3.366 

-    6-2-24.0 

-0.87 

-2. 68 

2.092 

1-557 

1   96 

800 

4030 

2.903 

3.605 

-16-5  -15.8 

-1  .17 

-1.72 

2.085       1.573 

1  .91 

1000 

6150 

3  .000 

3.789 

-15.7  -11.1 

-1  .14 

-1  .06 

2.086       1.583 

1    89 

1200 
1600 

8680 
14370 

3  .079 

3.938 

-18-7 
-26.9 

-6.0 

-2.02 

-0.55 
-0.18 

2-067 

1.5912 

1.82 

1  .73 

3.204 

4.157 

-2.0 

1.5971 

2000 

21000 

3  .301 

4.322 

0.0 

0.00 

1  .6000 

1  .67 

2500 

30300 

3  .398 

4.481 

+    0.9 

— 

+  0.C7 

1.6011 

1.62 

3000 
4000 

40500 
63400 

3  .477 

4.607 

+    1  .2 
—       -    0.2 

+  0.09 

1  .6014 

1.53 
1.58 

3.602 

4.802 

-0.02 

1  .5997 

5000 

9060C 

3  .699 

4.957 

—       -    0-2 

— 

-0.02 

— 

1  .5997 

1.59 

6000 

120600 

3.778 

5  .082 

-    0.9 

— 

-0.07 

—          1.5989 

1.61 

8000 
10000 

194100 
282500 

3  -903 

5.288 

+    0.7 

+  005 
+  0.17 

1  .6008 

1.66 

1  .77 

4.000 

5.451 

+    2.6 

1  .6027 

12000 

397500 

4.079 

5.599 

—       +    7.9 

— 

+  0-50 

1 .6080 

2.32 

140C0 

609500 

4.146 

5.785 

—       +27.7 

— 

+  1  .67 

1 .6267 

2   88 

16000 

907500 

4   204 

5. 958 

—       +45. 9 

+  2   85 

1    6456 

— 

^5=    9.459 

7.626 

4  = 

11.982 

12.945 

2-523 

5.319 

5. 319 

m=   — —  = 

=    2.11 

2.523 

-10  =  21.441 

20.571 

2.11X21.441  = 

=45.241 

J=- 

-24.670                                logn>=7.533 

+  2.11  log(B 

--10  = 

-2.467 

=  7.533      =log«i                      "'  =  0.0034 

i(B 

"i 

=  0.00341 

-4  =  13.380 

17.567 

V     — 

-*- 

J  = 

14.982 

20.129 

=    1.602 

2.562 

2.562 

»2= = 

1.602 

-    1.599^1-60 

-s  =28.362 

37.696 

1.60X28.362 
J  = 

=  45.379 

log  w  =  9  .040-1 

-1 .60  tog  (B 

~  7.683 

-="8  = 

-  0.960 

=  9.040    =log<M                     w  =  0. 10961 

R1'6 

!!■ 

=  '0.109 

B 

EMPIRICAL  CURVES.  251 

to  the  ranges  where  the  logarithmic  curve  is  not  a  straight 
line,  are  given  in  Table  VIII  as 

Aci\      Acio      An\      Jn,2 

tti  '       a-2  '       ri\  '      n2  ' 

they  are  calculated  as  follows: 

Assuming  n  as  constant,       no,  then  a  is  not  constant,   =  a0, 
and  the  ratio: 

J<i     a 

—  = 1 

a     (U) 

is  the  correction  factor,  and  it  is: 

w=a($>no, 

hence: 

log  io=log  a+no  log  (55 
and 

log  a     log  "'  -no  log  35; 

thus: 

log         log  a     log  ao=log  //'  —log  «o  —  no  log  (55, 

"11 

and 


J  <i     a  -—  

-1  =  Adog  w  -log  a0  -n0  log  15—1.       .     (1) 

Assuming  a  as  constant,    -  <i{].  then  rc  is  not  constant,  =n0, 
and  the  ratio, 

J  /?     n 

n      «o 
is  the  correction  factor,  and  it  is 

to  =  a0(&n ; 
hence 

log  w=log  ao+n  log  (B, 


252  ENGINEERING  MATHEMATICS. 

and 


thus 


and 


n  log  (B  =  log  w  —  log  do", 

■    n       n  log  (B     log  w  —log  a0 
no     n0  log  (B         no  log  (B 


Jn     n  log  ?/'  —log  a0  —  n0  log  (B 

_1  =  _&_        fa    o       o     b        ...     (2) 

n     n0  n0  log  (B 


by  these  equations  (1)  and  (2)  the  correction  factors  in  columns 
5  to  8  of  Table  VIII  are  calculated,  by  using  for  a0  and  n0  the 
values  of  the  lower  range  curve,  in  columns  5  and  7,  and  the 
values  of  the  medium  range  curve,  in  columns  6  and  8. 

Thus,  for  instance,  at  (B  =  100(),  the  loss  can  be  calculated 
by  the  ('([nation, 

w=a1(Rn\ 

by  applying  to  a,\  the  correction  factor: 

—15.7  per  cent  at  constant:  nx  =2.11,  that  is, 

«i  =0.00341(1  -0.157)  =0.00287; 

or  by  applying  to  r\\  the  correction  factor: 

-1.14  per  cent  at  constant:  ax  =0.00341,  that  is, 

nx  =2.11(1  -0.0114)  =2.086. 

Or  the  loss  can  be  calculated  by  the  equation, 

w  =  a2®n2, 
by  applying  to  <ij  the  correction  factor: 


1  I.I  per  cent  al  constant:  n2  =  1.60,  that  is, 

o2  =  0. 1096(1  -0.111)  =0.0974; 


EMPIRICAL   CURVES.  253 

or  by  applying  to  n2  the  correction  factor: 

—  1.00  per  cent  at  constant:  a2 =0.1096,  that  is, 

n2  =  1.60(1 -0.0106)  =1.583; 

and  the  loss  may  thus  be  given  bv  either  of  the  four  ex- 
pressions: 

w?=0.00287(R211=0.00341(R2086=0.0974(B1-6=0.1096(B1-587. 

As  seen,  the  variation  of  the  exponent  n,  required  to  extend 
the  use  of  the  parabolic  equation  into  the  range  for  which  it 
does  not  strictly  apply  any  more,  is  much  less  than  the  varia- 
tion of  the  coefficient  a,  and  a  far  greater  accuracy  is  thus 
secured  by  considering  the  exponent  n  as  constant — 1.6  for 
medium  and  high  values  of  -—(Band  making  the  correction  in 
coffiecient  a,  outside  of  the  range  where  the  1.6th  power  law 
holds  rigidly. 

In  the  last  column  of  Table  VIII  is  recorded  the  ratio  of 

.     •  -f  log  w 

variation,  m  =  ~rr1     -,  as  the  averages  each  oi  two  successive 
J  log  &' 

values.  As  seen,  m  agrees  with  the  exponent  n  within  the 
two  ranges,  where  it  is  constant,  but  differs  from  it  outside 
of  these  ranges.  For  instance,  if  B  changes  from  L600  down- 
ward, the  ratio  of  variation  m  increases,  while  the  exponent 
n  slightly  decreas<  • 

In  Fig.  83a  are  shown  the  percentage  correction  of  the 
coefficients  a,\  and  a2,  and  also  the  two  exponents  n{  and  n2, 
together  with  the  ratio  of  variation  m. 

The  ratio  of  variation  m  is  very  useful  in  calculating  the 
change  of  loss  resulting  from  a  small  change  of  magnetic  density, 
as  the  percentual  change  of  loss  w  is  m  times  the  percent  nal 
(small)  change  of  density. 

As  further  example,  the  reader  may  reduce  to  empirical 
equations  the  series  of  observations  given  in  Table  IX.  This 
table  gives: 

A.  The  candle-power  L,  as  function  of  the  power  input  p, 
of  a  40-watt  tungsten  filament  incandescent  lamp. 

B.  The  loss  of  power  by  corona  (discharge  into  the  air),  p, 
in  kw.,  in  1.895  km.  of  conductor,  as  function  of  the  voltage 
e  (in  kv.)   between  conductor  and  return  conductor,  for  the 


254 


ENGINEERING    MATHEMATICS. 


Table  IX. 


A.     Luminosity  characteristic  of  40-watt  tungsten  incandescent  lamp. 

L  =  horizontaI  candlepower. 
p=watts  input. 


2 

4 

8 

12 

16 


p 

L 

P 

L 

P 

L 

12.25 

20 

31.64 

40 

44.14 

128 

16.33 

24 

34.55 

44 

45-42 

192 

21.35 

28 

37.29 

48 

47.05 

256 

25.60 

32 

39  .26 

64 

54  .31 

291 

28.91 

30 

41  .47 

96 

65.73 

320 

1 

7677 
95.24 
109.0 
1182 

123 


382 
460 


135.6 
145.2 


1 


li.     Corona  less  of  high-voltage  transmission  lino;  at  60  cycles: 
1895  ni  .  length  of  conductor. 
3.10  m.  distance  between  conductors. 
No.  000  seven-strand  cable,  1.18  cm.  diameter. 
— 13°  C.;   76  .2  cm.  barometer;  sunshine. 
f=kilovoIts  between  conductors,  effective. 
p=  kilowatts  loss. 


79  8 
90.7 
101.5 
109.5 
120.5 
130.0 


V 

e 

P 

e- 

P 

e 

0.01 

141.5 

0.09 

181  .0 

1.02 

221.0 

0.01 

147.0 

0.08 

186.2 

1.55 

227.0 

0.02 

153.6 

0-12 

192.6 

2.49 

234.0 

0.03 

159.0 

0.16 

200.6 

3.77 

189  .0 

0.04 

169.8 

0.35 

208.6 

5  .34 

195.0 

0-06 

174.0 

0.53 

216.0 

7-13 

203.8 

1 

P 

e 

8-70 

212.0 

10-66 

219.0 

13.25 

— 

2-10 

— 

2.88 

— 

4-72 

6.44 

8-31 


Volume-pressure  characteristic  of  dry  steam  at  its  boiling-point. 

<= degrees  C. 
P  =  pressure,  in  kg.  per  cm.2 
V  =  volume,  in  m.3  per  kg. 


i 


59.8 

80.9 

99.1 

119.6 


p 

V 
7.806 

t 

P 

V 

t 

0.2 

132.8 

3.0 

0.612 

169.5 

0.5 

3.297 

142.8 

4-0 

0.467 

178.9 

1  .0 

1.717 

151.0 

5.0 

0.379 

186  9 

2.0 

0.896 

157.9 

6.0 

0.319 

197.2 

8-0 
10.0 
12.0 
15.0 


0.244 
0.197 
0.167 
0.135 


EMPIRICAL   CURVES.  255 

distance  of  310  cm.  between  the  conductors,  and  the  conductor 
diameter  of  1  .  L8  cm. 

C.  The  relation  between  steam  pressure  P,  in  kg.  per  cm.2, 
and  the  steam  volume  V,  in  m.3,  at  the  boiling-point,  per  kg. 
of  dry  steam. 

D.  Periodic  Curves. 

158.  All  periodic  functions  can  be  expressed  by  a  trigo- 
nometric series,  or  Fourier  series,  as  has  been  discussed  in 
Chapter  III,  and  the  methods  of  resolution  and  the  arrangements 
devised  to  carry  out  the  work  rapidly  have  also  been  dis- 
cussed in  Chapter  III. 

The  resolution  of  a  periodic  function  thus  consists  in  the 
determination  of  the  higher  harmonics,  which  are  super- 
imposed on  the  fundamental  wave. 

As  periodic  curves  are  of  the  greatest  importance  in  elec- 
trical engineering,  in  the  theory  of  alternating-current  phe- 
nomena, a  familiarity  with  the  wave  shapes  produced  by  the 
different  harmonics  is  desirable.  This  familiarity  should  be 
sufficient  to  enable  one  to  judge  immediately  from  the  shape 
of  the  wave,  as  given  by  oscillograph,  etc.,  which  harmonics 
are  present. 

The  effect  of  the  lower  harmonics,  such  as  the  third,  fifth, 
seventh,  etc.  (or  the  second,  fourth,  etc.,  where  present),  is 
to  change  the  shape  of  the  wave,  make  it  differ  from  sine 
shape,  and  in  the  "  Theory  and  Calculation  of  Alternating- 
current  Phenomena,"  4th.  Ed.,  Chapter  XXX,  a  number  of 
characteristic  distortions,  such  as  the  flat  top,  peaked  wave,  saw 
tooth,  double  and  triple  peaked,  sharp  zero,  flat  zero,  etc.,  have 
been  discussed  with  regard  to  the  harmonics  that  enter  into 
their  composition. 

159.  High  harmonics  do  not  change  the  shape  of  the  wave 
much,  but  superimpose  ripples  on  it,  and  by  counting  the 
number  of  ripples  per  half  wave,  or  per  wave,  the  order  of  the 
harmonic  can  rapidly  be  determined.  For  instance,  the  wave 
shown  in  Fig.  84  contains  mainly  the  eleventh  harmonic,  as 
there  are  eleven  ripples  per  wave  (Fig.  84). 

Very  frequently  high  harmonics  appear  in  pairs  of  nearly 
the  same  frequency  and  intensity,  as  an  eleventh  and  a  thir- 


2m 


ENGINEERING   MATHEMATK  'S. 


teenth  harmonic,  etc.  In  this  case,  the  ripples  in  the  wave 
shape  show  maxima,  where  the  two  harmonics  coincide,  and 
nodes,  where  the  two  harmonics  are  in  opposition.  The 
presence  of  nodes  makes  the  counting  of  the  number  of  ripples 
per  complete  wave  more  difficult.  A  convenient  method  of 
procedure  in  this  case  is,  to  measure  the  distance  or  space 


Fig.  84.     Wave  in  which  Eleventh  Harmonic  Predominates. 

between  the  maxima  of  one  or  a  few  ripples  in  the  .range  where 
they  are  pronounced,  and  count  the  number  of  nodes  per 
cycle.  For  instance,  in  the  wave,  Fig.  85,  the  space  of  two 
ripples  is  about  GO  deg.,  and  two  nodes  exist  per  complete 

Qf*f\ 

wave.     60  deg.  for  two  ripples,  gives  2X-— =12  ripples  per 


Fig.  85.     Wave  in  which  Eleventh  .and  Thirteenth  Harmonics  Predominate. 

complete  wave,  as  the  average  frequency  of  the  two  existing 
harmonics,  and  since  these  harmonics  differ  by  2  (since  there 
are  two  nodes),  their  order  is  the  eleventh  and  the  thirteenth 
harmonics. 

This  method  of  determining  two  similar  harmonics,  with  a 
little  practice,  becomes  very  convenient  and  useful,  and   may 


EMPIRICAL   CURVES.  257 

frequently  be  used  visually  also,  in  determining  the  frequency 
of  hunting  of  synchronous  machines,  etc.  In  the  phenomenon 
of  hunting,  frequently  two  periods  are  superimposed,  a  forced 
frequency,  resulting  from  the  speed  of  generator,  etc.,  and  the 
natural  frequency  of  the  machine.  Counting  the  number  of 
impulses,  /,  per  minute,  and  the  number  of  nodes,  n,  gives  the 

Tb  7b 

two  frequencies :/+- and/—-;  and  as  one  of  these  frequencies 

is  the  impressed  engine  frequency,  this  affords  a  check. 

Not  infrequently  wave-shape  distortions  are  met,  which 
are  not  due  to  higher  harmonics  of  the  fundamental  wave, 
but  are  incommensurable  therewith.  In  this  case  there  are 
two  entirely  unrelated  frequencies.  This,  for  instance,  occurs 
in  the  secondary  circuit  of  the  single-phase  induction  motor; 
two  sets  of  currents,  of  the  frequencies  fs  and  (2f—fs)  exist 
(where  /  is  the  primary  frequency  and  fs  the  frequency  of 
slip).  Of  this  nature,  frequently,  is  the  distortion  produced  by 
surges,  oscillations,  arcing  grounds,  etc.,  in  electric  circuits; 
it  is  a  combination  of  the  natural  frequency  of  the  circuit 
with  the  impressed  frequency.  Telephonic  currents  commonly 
show  such  multiple  frequencies,  which  are  not  harmonics  of 
each  other. 


CHAPTER   VII. 
NUMERICAL   CALCULATIONS. 

160.  Engineering  work  leads  to  more  or  less  extensive 
numerical  calculations,  when  applying  the  general  theoretical 
investigation  to  the  specific  cases  which  are  under  considera- 
tion.    Of  importance  in  such  engineering  calculations  are: 

(a)  The  method  of  calculation. 

(&)  The  degree  of  exactness  required  in  the  calculation. 

(c)  The  intelligibility  of  the  results. 

(d)  The  reliability  of  the  calculation. 

a.  Method  of  Calculation. 

Before  beginning  a  more  extensive  calculation,  it  is  desirable 
carefully  to  scrutinize  and  to  investigate  the  method,  to  find 
the  simplest  way,  since  frequently  by  a  suitable  method  and 
system  of  calculation  the  work  can  be  reduced  to  a  small  frac- 
tion of  what  it  would  otherwise  be,  and  what  appear  to  be 
hopelessly  complex  calculations  may  thus  be  carried  out 
quickly  and  expeditiously  by  a  proper  arrangement  of  the 
work.  Indeed ,  the  most  important  part  of  engineering  work — and 
also  of  other  scientific  work — is  the  determination  of  the  method 
of  attacking  the  problem,  whatever  it  may  be,  whether  an 
experimental  investigation,  or  a  theoretical  calculation.  It  is 
very  rarely  that  important  problems  can  be  solved  by  a  direct 
attack,  by  brutally  forcing  a  solution,  and  then  only  by  wasting 
a  large  amount  of  work  unnecessarily.  It  is  by  the  choice  of 
a  suitable  method  of  attack,  that  intricate  problems  are  reduced 
to  simple  phenomena,  and  then  easily  solved:  frequently  in 
such  cases  requiring  no  solution  at  all,  but  being  obvious  when 
looked  at  from  the  proper  viewpoint. 

Before  attacking  a  more  complicated  problem  experimentally 
or  theoretically,  considerable  time  and  study  should  thus  first  be 
devoted  to  the  determination  of  a  suitable  method  of  attack. 

25S 


NUMERICAL  CALCULATIONS.  259 

The  next  then,  in  cases  whore  considerable  numerical  calcu- 
lations are  required,  is  the  method  of  calculation.  The  most 
convenient  one  usually  is  the  arrangement  in  tabular  form. 

As  example,  consider  the  problem  of  calculating  the  regula- 
tion of  a  00,000- volt  transmission  line,  of  r  =  60  ohms  resist- 
ance, z  =  135  ohms  inductive  reactance,  and  5=0.0012  conden- 
sive  susceptance,  for  various  values  of  non-inductive,  inductive, 
and  condensive  load. 

Starting  with  the  complete  equations  of  the  long-distance 
transmission  line,  as  given  in  "  Theory  and  Calculation  of 
Transient  Electric  Phenomena  and  Oscillations,"  Section  III, 
paragraph  0,  and  considering  that  for  every  one  of  the  various 
power-factors,  lag,  and  lead,  a  sufficient  number  of  values 
have  to  be  calculated  to  give  a  curve,  the  amount  of  work 
appears  hopelessly  large. 

However,  without  loss  of  engineering  exactness,  the  equa- 
tion of  the  transmission  line  can  be  simplified  by  approxima- 
tion, as  discussed  in  Chapter  V,  paragraph  123,  to  the  form, 


f        7Y 1  7Y\ 

B1_g0Jl+_j+Zfc(l+-g-J; 


Wofl+4^     +W.fl+T 


(1) 


where  Eq,  Jo  are  voltage  and  current,  respectively  at  the  step- 
down  end,  Ei,  1\  at  the  sitep-up  end  of  the  line;  and 

Z  =  r— /.r  =  60  — 135/  is  the  total  line  impedance; 

Y  =  g  —  jb=  —0.0012/  is  the  total  shunted  line  admittance. 

Herefrom  follow  the  numerical  values: 

ZY  (60- 1357)  (-0.0012/) 

1+-2-1+ -> 

=  1-0.036/-  0.081  =0.919-0.036/; 

ZY 

1  +-tt  =  1  -0.012?  -  0.027  =  0.973  -  0.012/; 
6 


260 


ENGINEERING   MATH  KM  A  TICS. 


(60- 135?)  (0.973 -0.012/) 

=  58. 1-0.72/- 131.1/- 1.62  =  56.8- 131.8/; 

=  (-0.0012/)(0.973-0.012/) 

=  -0.001168/-0.0000144  =  (-0.0144-1.168/)10-5 


hence,   substituting   in   (1),   the   following  equations   may   to 
written: 


#!  =  (0.919-0.036/)#o +  (56.8-131.8/)/0  =  ^  \  />'• 
/i  =  (0.919-0.036/)7o  -  (0.0144  +1. L(iS/)/snl()  -  =  C- />.  /  [') 
161.  Now   the   work  of  calculating   a  .scries   of  numerical 
values  is  continued  in  tabular  form,  as  follows: 

1.    100    PER   CENT    POWEK-F ACTOR. 

£■0=60  kv.  at  step-down  end  of  line. 

A  =  (0.919 -O.036/)£'o=  55. 1-2.2/  kv. 

D=  (0.0144+  1.168/tEo  10-3  =  0.9  +  70.1j  amp. 


r 
/0  amp. 

B  kv. 

Ei  =  ei-je2 

=  A+B. 

n2  +  C22=e2. 

e 
55.1 

—  =  tanc. 
ei 

4e. 

0 

0 

55.1-    2.2/ 

3036+      5  =  3041 

-  0 . 040 

-    2.3 

20 

1.1-    2.6/ 

56.2-    4.8/ 

3158+    23  =  3181 

56.4 

-0.085 

-    4  9 

40 

2.3-    5.3/ 

57.4-    9.5/ 

3295+    56  =  3351 

57.9 

-0.131 

-    7.5 

60 

3.4-    7.9; 

58.5-10.1/ 

3422+102  =  3524 

59.4 

-0.173 

-    9.9 

80 

4.5-10.5/ 

59.6-12.7/ 

3552  +  161  =  3713 

60.9 

-0.213 

-12.0 

100 

5.7-13.2/ 

60.8-15.4/ 

3697  +  237  =  3934 

62.7 

-0.253 

-14.2 

120 

6.8-15.8/ 

61.9-18.0/ 

3832  +  324=4156 

64.5 

-0.291 

-16.3 

/o 
amp. 

C  amp. 

Ji  =  u—  jit 
=  C-D 

ii2  +  i22  =  i2 

i 

it 

—  =  tam' 

ii 

A-i 

i  4-e= 

2$_e,  i 

Power- 
factor 

0 

0 

-0.7-90.1/ 

4914+1          =    4915 

70.1 

+  78 

+  89.1 
-90.9 

-88.6 

0.024 

20 

18.4-0.7; 

17.5-70.8/ 

5013+   306=    5319 

72.9 

-4.04 

-76.3 

-71.4 

0.332 

40 

36.8-1.4/ 

35.9-71.5/ 

5112  +  1289=    6401 

80.0 

-1.99 

-63.4 

-55.9 

0  558 

60 

55.1-2.2; 

54.2-72.3/ 

5227  +  2938=    8165 

90.4 

-1.33 

-53.1 

-43.2 

0  728 

80 

73.5-2.9/ 

72.6-73.0; 

5329  +  5271=10600 

103.0 

-1.055 

-45.2 

-33.2 

0  837 

100 

91.9-3.6/ 

91.0-73.9/ 

8281  +  5432=13713 

117.1 

-0.811 

-39.1 

-24.9 

0  907 

120 

110.3-4.3/ 

109.4-74.4; 

11969  +  5535=17504 

132.3 

-0.680 

-34.1 

-17.8 

0  952 
lead 

NUMERICAL  CALCULATIONS. 


261 


ei=  60  kv.  at 

step-up  end  of  line. 

h 

amp. 

Red.  Factor, 

c 
60 

airi]>. 

kv. 

u 
amp. 

Power-Factor. 

0 

0  918 

0 

65.5 

76.4 

0  024 

20 

0  940 

21.3 

63.8 

77.5 

0.332 

40 

0.965 

41  4 

62.1 

82.9 

0.558 

60 

0.990 

60   6 

60.6 

91.4 

0.728 

80 

1.015 

78   8 

59.1 

101.5 

0  837 

100 

1  045 

95.7 

57.5 

112  3 

0  907 

120 

1.075 

111.7 

55.8 

122  8 

0.952 
lead 

Curv 

ea  of  <„.  >„.  /,. 

I,  plotted  in  Fig.  86. 

- 

1 

2.  90  Per  Cent  Power-Factor,  Lag. 


cos  0=0.9;    sin  0     \  1     0.92=0.436; 
/o='n  cos  0+j sin  0)=io  0.9  1-0.436/  ; 

Ei     f0.919-0.036/)e0  +  (56.8- 131. 8/   0.9  r0.436/)io 

(0.919-0.036/)e0  +  (108.5-  93.8     0=A+B'\ 
7i  =  (0.919-0.036j)(0.9+0.436j)io-  (0.0144 +1.168j>ol0"3 
=  (0.843 +0.366/HV   (0.0144+1.168j>010-3=C/-D< 

and  now  the  table  is  calculated  in  the  same  manner  as  under  1. 
Then    corresponding   tables    arc    calculated,    in    the   same 
manner,   for  power-factor,    =0.8  and    =0.7,   respectively,   lag, 
and  for  power-factor  =0.9,  O.s,  0.7,  lead:  that  is,  for 

cos  0+/sin  0=0.8  H  0.6/; 

0.7     0.714./; 
0.9-0.436/; 
us  -0.6/; 
0.7-0.711  . 

Then  curves  arc  plotted  for  all  seven  values  of  power-factor, 
from  0.7  lag  to  0.7  lead. 

From  these  curves,  for  a  number  of  values  of  l*o,  for  instance, 
io  =  20,  40,  60,  SO,  100,  numerical  values  of  i\,  e0,  cos  0,  are 


2G2 


ENGINEERING  MA  Til  EM  A  TICS. 


taken,  and  plotted  as  curves,  which,  for  the  same  voltage 
ei  =  60  at  the  step-up  end,  give  ih  e0,  and  cos  0,  for  the  same 
value  %q,  that  is,  give  the  regulation  of  the  line  at  constant 
current  output  for  varying  power-factor. 


b.  Accuracy  of  Calculation. 

162.  Not  all  engineering  calculations  require  the  same 
degree  of  accuracy.  When  calculating  the  efficiency  of  a  large 
alternator  it  may  be  of  importance  to  determine  whether  it  is 
97.7  or  97.8  per  cent,  that  is,  an  accuracy  within  one-tenth 
per  cent  may  be  required;  in  other  cases,  as  for  instance, 
when  estimating  the  voltage  which  may  be  produced  in  an 
electric  circuit  by  a  line  disturbance,  it  may  be  sufficient  to 


1001.00 


80  0.80 


0.60 


0.40 


0.20 


Fig.  86.     Transmission  Line  Characteristics. 

determine  whether  this  voltage  would  be  limited  to  double 
the  normal  circuit  voltage,  or  whether  it  might  be  5  or  10 
times  the  normal  voltage. 

In  general,  according  to  the  degree  of  accuracy,  engineering 
calculations  may  be  roughly  divided  into  three  classes: 


NUMERICAL   CALCULATIONS.  263 

(a)  Estimation  of  the  magnitude  of  an  effect;  that  is, 
determining  approximate  numerical  values  within  25,  50,  or 
100  per  cent.  Very  frequently  such  very  rough  approximation 
is  sufficient,  and  is  all  that  can  be  expected  or  calculated. 
For  instance,  when  investigating  the  short-circuit  current  of  an 
electric  generating  system,  it  is  of  importance  to  know  whether 
this  current  is  3  or  4  times  normal  current,  or  whether  it  is 
40  to  50  times  normal  current,  but  it  is  immaterial  whether 
it  is  45  to  46  or  50  times  normal.  In  studying  lightning 
phenomena,  and,  in  general,  abnormal  voltages  in  electric 
systems,  calculating  the  discharge  capacity  of  lightning  arres- 
ters, etc.,  the  magnitude  of  the  quantity  is  often  sufficient.  In 
calculating  the  critical  speed  of  turbine  alternators,  or  the 
natural  period  of  oscillation  of  synchronous  machines,  the 
same  applies,  since  it  is  of  importance  only  to  see  that  these 
speeds  are  sufficiently  remote  from  the  normal  operating  speed 
to  give  no  trouble  in  operation. 

(b)  Approximate  calculation,  requiring  an  accuracy  of  one 
or  a  few  per  cent  only;  a  large  pail  of  engineering  calcu- 
lations fall  in  this  class,  especially  calculations  in  the  realm  of 
design.  Although,  frequently,  a  higher  accuracy  could  be 
reached  in  the  calculation  proper,  ii  would  lie  of  no  value, 
since  the  data  on  which  the  calculations  are  based  arc  sus- 
ceptible to  variations  beyond  control,  due  to  variation  in  the 
material,  in  the  mechanical  dimensions,  etc. 

Thus,  for  instance,  the  exciting  current  of  induction  motors 
may  vary  by  several  per  cent,  due  to  variations  of  the  length 
of  air  gap,  so  small  as  to  be  beyond  the  limits  of  constructive 
accuracy,  and  a  calculation  exact  to  a  fraction  of  one  per  cent, 
while  theoretically  possible,  thus  would  be  practically  useless, 
The  calculation  of  the  ampere-turns  required  for  the  shunt 
field  excitation,  or  for  the  series  field  of  a  direct-current 
generator  needs  only  moderate  exactness,  as  variations  in  the 
magnetic  material,  in  the  speed  regulation  of  the  driving 
power,  etc.,  produce  differences  amounting  to  several  per 
cent. 

(c)  Exact  engineering  calculations,  as,  for  instance,  the 
calculations  of  the  efficiency  of  apparatus,  the  regulation  of 
transformers,  the  characteristic  curves  of  induction  motors, 
etc.  These  are  determined  with  an  accuracy  frequently  amount- 
ing to  one-tenth  of  one  per  cent  and  even  greater. 


264  ENGINEERING  MATHEMATICS. 

Even  for  most  exact  engineering  calculations,  the  accuracy 
of  the  slide  rule  is  usually  sufficient,   if  intelligently  used,  that 
is,  used  so  as  to  get  the  greatest  accuracy.     For  accurate  calcu 
lations,  preferably  the  glass  slide  should  not  be  used,    but  the 
result  interpolated  by  the  eye. 

Thereby  an  accuracy  within  {  per  cent  can  easily  be  main- 
tained. 

For  most  engineering  calculations,  logarithmic  tables  are 
sufficient  for  three  decimals,  if  intelligently  used,  and  as  such 
tables  can  be  contained  on  a  single  page,  their  use  makes  the 
calculation  very  much  more  expeditious  than  tables  of  more 
decimals.  The  same  applies  to  trigonometric  tables:  tables 
of  the  trigonometric  functions  (not  their  logarithms)  of  three 
decimals  I  find  most  convenient  for  most  cases,  given  from 
degree  to  degree,  and  using  decimal  fractions  of  the  degrees 
(not  minutes  and  seconds).* 

Expedition  in  engineering  calculations  thus  requires  the  use 
of  tools  of  no  higher  accuracy  than  required  in  the  result,  and 
such  are  the  slide  rules,  and  the  three  decimal  logarithmic  and 
trigonometric  tables.  The  use  of  these,  however,  make  it 
neccessary  to  guard  in  the  calculation  against  a  loss  of  accuracy. 

Such  loss  of  accuracy  occurs  in  subtracting  or  dividing  two 
terms  which  are  nearly  equal,  in  some  logarithmic  operations, 
solution  of  equation,  etc,,  and  in  such  cases  either  a  higher 
accuracy  of  calculation  must  be  employed — seven  decimal 
logarithmic  tables,  etc. — or  the  operation,  which  lowers  the 
accuracy,  avoided.  The  latter  can  usually  be  done.  For 
instance,  in  dividing  297  by  283  by  the  slide  rule,  the  proper 
way  is  to  divide  297-283  =  14  by  283,  and  add  the  result 
to  1. 

It  is  in  the  methods  of  calculation  that  experience  and  judg- 
ment and  skill  in  efficiency  of  arrangement  of  numerical  calcu- 
lations is  most  marked. 

163.  While  the  calculations  are  unsatisfactory,  if  not  carried 
out  with  the  degree  of  exactness  which  is  feasible  and  desirable, 
it  is  equally  wrong  to  give  numerical  values  with  a  number  of 

*  This  obviously  does  not  apply  to  some  classes  of  engineering  work,  in 
which  a  much  higher  accuracy  of  trigonometric  functions  is  required,  as 
trigonometric  surveying,  etc. 


NUMEL'K  AL  CALCULATIONS.  265 

ciphers  greater  than  the  method  or  the  purpose  of  the  calcula- 
tion warrants.  For  instance,  if  in  the  design  of  a  direct-current 
generator,  the  calculated  field  ampere-turns  are  given  as  9738, 
such  a  numerical  value  destroys  the  confidence  in  the  work  of 
the  calculator  or  designer,  as  it  implies  an  accuracy  greater 
than  possible,  and  thereby  shows  a  lack  of  judgment. 

The  number  of  ciphers  in  which  the  result  of  calculation  is 
given  should  signify  the  exactness,  In  this  respect  two 
systems  are  in  use: 

(a)  Numerical  values  arc  given  with  one  more  decimal 
than  warranted  by  the  probable  error  of  the  result:  thai  i-. 
the  decimal  before  the  lasl  is  cornet,  hut  the  last  decimal  may 
be  wrong  by  several  units.  This  method  is  usually  employed 
in  astronomy,  physics,  etc. 

(&)  Numerical  values  arc  given  with  a.-  many  decimals  as 
the  accuracy  of  the  calculation  warrants;  that  is,  the  lasl 
decimal  is  probably  correel  within  half  a  unit.  For  instance, 
an  efficiency  of  86  per  cent  mean-  an  efficiency  between  85.5 
and  Si')..")  pei-  cent:  an  efficiency  of  97.3  per  cent  mean-  an 
efficiency  between  97.25  and  97.35  per  cent,  etc.  This  system 
is  generally  used  in  engineering  calculation-.  To  get  accuracy 
of  the  last  decimal  of  the  result,  the  calculations  then  nm-t 
he  carried  out  for  one  more  decimal  than  given  in  the  result. 
For  instance,  when  calculating  the  efficiency  by  adding  the 
various  percentages  of  losses,  data  like  the  following  may  be 
given; 

Core  loss 2.73  per  cent 

/-/• 1.06 


Friction 0.93 


a 


Total 4.72 

Efficiency 100-4.72  =  95,;s 

Approximately 0.1 1  " 

It  is  obvious  that  throughout   the    same  calculation  the 
same  degree  of  accuracy  must  be  observed. 
It  follows  herefrom  that  the  values: 

2\\    2.5;     2.50;     2.500, 


26<]  ENGINEERING  MATHEMATICS. 

while  mathematically  equal,  are  not  equal  in  their  meaning  as 
an  engineering  result : 

2.5      means  between  2.45    and  2.55; 
2.50    means  between  2.495    and  2.505; 
2.500  means  between  2.4995  and  2.5005; 

while  2\  gives  no  clue  to  the  accuracy  of  the  value. 

Thus  it  is  not  permissible  to  add  zeros,  or  drop  zeros  at 
the  end  of  numerical  values,  nor  is  it  permissible,  for  instance, 
to  replace  fractions  as  1/16  by  0.0625,  without  changing  the 
meaning  of  the  numerical  value,  as  regards  its  accuracy. 
This  is  not  always  realized,  and  especially  in  the  reduction  of 
common  fractions  to  decimals  an  unjustified  laxness  exists 
which  impairs  the  reliability  of  the  results.  For  instance,  if 
in  an  arc  lamp  the  arc  length,  for  which  the  mechanism  is 
adjusted,  is  stated  to  be  0.8125  inch,  such  a  statement  is 
ridiculous,  as  no  arc  lamp  mechanism  can  control  for  one-tenth 
as  great  an  accuracy  as  implied  in  this  numerical  value:  the 
value  is  an  unjustified  translation  from  13/16  inch. 

The  principle  thus  should  be  adhered  to,  that  all  calcula- 
tions are  carried  out  for  one  decimal  more  than  the  exactness 
required  or  feasible,  and  in  the  result  the  last  decimal  dropped; 
that  is,  the  result  given  so  that  the  last  decimal  is  probably 
correct  within  half  a  unit. 

c.  Intelligibility  of  Engineering  Data. 

164.  In  engineering  calculations  the  value  of  the  results 
mainly  depends  on  the  information  derived  from  them,  that  is, 
on  their  intelligibility.  To  make  the  numerical  results  and 
their  meaning  as  intelligible  as  possible,  it  thus  is  desirable, 
whenever  a  series  of  values  are  calculated,  to  carefully  arrange 
them  in  tables  and  plot  them  in  a  curve  or  in  curves.  The 
latter  is  necessary,  since  for  most  engineers  the  plotted  curve 
gives  a  much  better  conception  of  the  shape  and  the  variation 
of  a  quantity  than  numerical  tables. 

Even  where  only  a  single  point  is  required,  as  the  core 
loss  at  full  load,  or  the  excitation  of  an  electric  generator  at 
rated  voltage,   it   is  generally  preferable  to  calculate  a  few 


NUMERICAL  CALCULATIONS. 


267 


Volts 

■566 

^•iUU 

oUU 

-200 

K100 

0 

2 

0 

I 

0 

____ 

6 

0 

8 

1 

0 

Fig.  87.     <  Compounding  <  Curve. 


points  near  the  desired  value,  so  as  to  get  at  least  a  short  piece 
of  curve  including  the  desired  point. 

The  main  advantage,  and  foremost  purpose  of  curve  plotting 
thus  is  to  show  the  shape  of  the  function,  and  thereby  give 
a  clearer  conception  of  it  ; 
hut  for  recording  numerical 
values,  and  deriving  numer- 
ical values  from  it ,  the  plotted 
curve  is  inferior  to  the  table, 
due  to  the  limited  accuracy 
possible  'm  a  plotted  curve, 
and  the  further  inaccuracy 
resulting  when  drawing  a 
curve  through  the  plotted  cal- 
culated points.  To  some 
extent ,  the  numerical  values 
a-  taken  from  a  plol  ted  curve, 
depend  on  the  particular 
kind  of  curve  rule  used  in 
plotting  the  curve. 

In  general,  curves  are  used  for  two  different  purposes,  ami 

on  the  purpose    for    which    the   curve  i<   plotted,  should  depend 

the  method  of  plotting,  as  the  scale,  the  zero  values,  etc. 

When  curves  are  used  to 

illustrate  the  shape  of  the 
function,  so  as  to  show  how 
much  and  in  what  manner  a 
quantity  varies  as  fund  inn 
of  and  her,  large  divisions  of 
inconspicuous  cross-seel  ion- 
ing  are  desirable,  hut  it  is 
49o]  essential  that  the  cross- 
sectioning  should  extend  to 
the  zero  value-  of  the  func- 
tion, even  if  the  numerical 
values  do  not  extend  so 
far,  since  otherwise  a  wrong 
impression  would  he  con- 
ferred. As  illustrations  are  plotted  in  Figs.  87  and  88,  the 
compounding  curve  of  a  direct-current  generator.     The  arrange- 


V 

-530 

•»u 

^111 

-4JU 

^itO 

0 

2 

0 

4 

0 

6 

o 

8 

1 

0 

Fig.  ss.     Compounding  Curve. 


268 


ENGINEERING  MATHEMATICS. 


incut  in  Fig.  87  is  correct  ;  it  shows  the  relative  variation 
of  voltage  as  function  of  the  load.  Fig.  88,  in  which  the 
cross-sectioning  docs  not  begin  at   the  scale   zero,  confers  the 


___ 

SI 

1£ 

> 

X 

1 

0 

2 

i 

3 

0 

4 

3 

5 

0 

60 

7 

0 

Fig.  SO.     Curve  Plotted  to  show  Characteristic  Shape. 


rr—  - 

5t'::::  : 

U-  mi 
Ft  ::':: 

^ 

' 

— 

'■" 



r~ — 

88 

I 

:_: 

— 

:::: 
1 

■ 
— 

■' 

§ 

LJ 

I: 

,:: 
-:   ^ 

m 

•::1 

).: 

[IV: 
■1 

^ 

- 

) 

,1 

— 

~.i> 

)    . 

— 

(i 

— 1 
:::: 

' 

J  : 

— i 
if 

m 

1 

n 

#1 

Fig.  90.     Curve  Plotted  for  Use  as  Design  Data. 

wrong  impression  that  the  variation  of  voltage  is  far  greater 
than  it  really  is. 

When   curves    are    used    to   record     numerical   values   and 
derive  them  from  the  curve,  as,  for  instance,  is  commonly  the 


NUMERICAL  CALCULATIONS. 


269 


case  with  magnetizal  ion  curves,  it  is  unnecessary  to  have  the  zero 
of  the  function  coincide  with  the  zero  of  the  cross-sectioning,  bu1 
rather  preferable  not  to  have  it  so,  if  thereby  a  better  scale  of 
the  curve  can  be  secured,  li  is  desirable,  however,  to  use  suffi- 
ciently small  cross-sectioning  to  make  it  possible  to  take 
numerical  values  from  the  curve  with  good  accuracy.  This  is 
illustrated  by  Figs.  89and  90.  Both  show  the  magnetic  charac- 
teristic of  sofl  steel,  for  the  range  above  <B  =  .S000,  in  which  it  is 
usually  employed.  Fig.  89  show,-  the  proper  way  of  plotting 
for  showing  the  shape  of  the  function.  Fig.  90  the  proper  way  of 
plotting  for  use  of  the  curve  to  derive  numerical  values  therefrom. 


^ 

II 

-Ln 

Fig.  91.     Same  Function  Plotted  to  Different  scale-;   I  is  correct. 

165.  Curves  should  be  plotted  in  such  a  manner  as  to  show 
the  quantity  which  they  represent,  and  its  variation,  a-  well  as 
possible.    Two  features  are  desirable  herefor: 

1.  To  use  such  a  scale  that  the  average  slope  of  the  curve, 
or  at  least  of  the  more  'important  part  of  it,  does  no1  differ 
much  from  45  deg.  Bereby  variations  of  curvature  are  besl 
shown.  To  illustrate  this,  the  exponential  function  y=e~x  is 
plotted  in  three  different  scales,  as  curves  I,  II,  III,  in  Fig.  91. 
Curve  I  has  the  proper  scale. 

2.  To  use  such  a  scale,  that  the  total  range  of  ordinates  is 
not  much  different  from  the  total  range  of  abscissas.  Thus 
when  plotting  the  power-factor  of  an  induction  motor,  in 
Fig.  92,  curve  I  is  preferable  to  curves  II  or  III. 


270 


ENGINEERING  MA  THEM  A  TICS. 


Those  two  requirements  frequently  are  at  variance  with 
each  other,  and  then  a  compromise  has  to  be  made  between 
them,  that  is,  such  a  scale  chosen  that  the  total  ranges  of  the 
two  coordinates  do  not  differ  much,  and  at  the  same  time 
the  average  slope  of  the  curve  is  not  far  from  45  (leg.  This 
usually  leads  to  a  somewhat  rectangular  area  covered  by  the 
curve,  as  shown,  for  instance,  by  curve  I,  in  Fig.  91. 

In  curve  plotting,  a  scale  should  be  used  which  is  easily 
read.  Hence,  only  full  scale,  double  scale,  and  half  scale 
should  be  used.  Triple  scale  and  one-third  scale  are  practically 
unreadable,  and  should  therefore  never  be  used.     Quadruple 


11/ 

I 

/ 

III 

■>! 

// 

// 

V 



Fig.  92.     Same  Function  Plotted  to  Different  Scales;  I  is  Correct. 


scale  and  quarter  scale  are  difficult  to  read  and  therefore  unde- 
sirable, and  are  generally  unnecessary,  since  quadruple  scale 
is  not  much  different  from  half  scale  with  a  ten  times  smaller 
unit,  and  quarter  scale  not  much  different  from  double  scale 
of  a  ten  times  larger  unit. 

166.  Any  engineering  calculation  on  which  it  is  worth 
while  to  devote  any  time,  is  worth  being  recorded  with  suffi- 
cient completeness  to  be  generally  intelligible.  Very  often  in 
making  calculations  the  data  on  which  the  calculation  is  based, 
the  subject  and  the  purpose  of  the  calculation  are  given  incom- 
pletely or  not  at  all,  since  they  are  familiar  to  the  calculator  at 
the  time  of  calculation.     The  calculation  thus  would  be  unin- 


NUMERICAL  CALCULATIONS.  271 

telligible  to  any  other  engineer,  and  usually  becomes  unintelli- 
gible even  to  the  calculator  in  a  few  week-. 

In  addition  to  the  name  and  the  date,  all  calculations  should 
be  accompanied  by  a  complete  record  of  the  object  and  purpose 
of  the  calculation,  the  apparatus,  the  assumptions  made,  the 
data  used,  reference  to  other  calculations  or  data  employed, 
etc.,  in  short,  they  should  include  all  the  information  required 
to  make  the  calculation  intelligible  to  another  engineer  without 
fiui her  information  besides  that  contained  in  the  calculations, 
or  in  the  references  given  therein.  The  small  amount  of  time 
and  work  required  to  do  this  is  negligible  compared  with  the 
increased  utility  of  the  calculation. 

Tables  and  curve-  belonging  to  the  calculation  should  in 
the  same  way  he  completely  identified  with  it  and  contain 
sufficient  data  to  he  intelligible. 

d.   Reliability  of  Numerical  Calculations. 

167.  The  most  important  and  essential  requirement  of 
numerical  engineering  calculations  is  their  absolute  reliability. 
When  making  a  calculation,  the  most  brilliant  ability,  theo- 
retical knowledge  and  practical  experience  of  an  engineer  are 
made  useless,  and  even  worse  than  useless,  by  a  single  error  in 
an  important  calculat i<>n. 

Reliability  of  the  numerical  calculation  is  of  vastly  greater 
importance  in  engineering  than  in  any  other  field.  In  pure 
mathematics  an  error  in  the  numerical  calculation  of  an 
example  which  illustrates  ;i  general  proposit  ion,  doe-  not  detract 
from  the  interest  and  value  of  the  latter,  which  is  the  main 
purpose;  in  physics,  the  general  law  which  is  the  subject  of 
the  investigation  remains  true,  and  the  investigation  of  interest 
and  use,  even  if  in  the  numerical  illustration  of  the  law  an 
error  i^  made.  With  the  most  brilliant  engineering  design, 
however,  if  in  the  numerical  calculation  of  a  single  structural 
member  an  error  has  been  made,  and  its  strength  thereby  calcu- 
lated wrong,  the  rotor  oi  the  machine  (lies  to  pieces  by  centrifugal 
forces,  or  the  bridge  collapses,  and  with  it  the  reputation  of  the 
engineer.  The  essential  difference  between  engineering  and 
purely  scientific  caclulations  is  the  rapid  check  on  the  correct- 
ness of  the  calculation,  which  is  usually  afforded  by  the  per- 


272  ENGINEERING  MATHEMATICS. 

formance  of  the  calculated  structure — but  too  late  to  correct 
errors. 

Thus  rapidity  of  calculation,  while  by  itself  useful,  is  of  no 
value  whatever  compared  with  reliability — that  is,  correct- 
ness. 

One  of  the  first  and  most  important  requirements  to  secure 
reliability  is  neatness  and  care  in  the  execution  of  the  calcula- 
tion. If  the  calculation  is  made  on  any  kind  of  a  sheet  of 
paper,  with  lead  pencil,  with  frequent  striking  out  and  correct- 
ing of  figures,  etc.,  it  is  practically  hopeless  to  expect  correct 
results  from  any  more  extensive  calculations.  Thus  the  work 
should  be  done  with  pen  and  ink,  on  white  ruled  paper;  if 
changes  have  to  be  made,  they  should  preferably  be  made  by 
erasing,  and  not  by  striking  out.  In  general,  the  appearance1  of 
the  work  is  one  of  the  best  indications  of  its  reliability.  The 
arrangement  in  tabular  form,  where  a  scries  of  values  are  calcu- 
lated, offers  considerable  assistance  in  improving  the  reliability. 

168.  Essential  in  all  extensive  calculations  is  a  complete 
system  of  checking  the  results,  to  insure  correctness. 

One  way  is  to  have  the  same  calculation  made  independently 
by  two  different  calculators,  and  then  compare  the  results. 
Another  way  is  to  have  a  few  points  of  the  calculation  checked 
by  somebody  else.  Neither  way  is  satisfactory,  as  it  is  not 
always  possible  for  an  engineer  to  have  the  assistance  of  another 
engineer  to  check  his  work,  and  besides  this,  an  engineer  should 
and  must  be  able  to  make  numerical  calculations  so  that  he  can 
absolutely  rely  on  their  correctness  without  somebody  else 
assisting  him. 

In  any  more  important  calculations  every  operation  thus 
should  be  performed  twice,  preferably  in  a  different  manner. 
Thus,  when  multiplying  or  dividing  by  the  slide  rule,  the  multi- 
plication or  division  should  be  repeated  mentally,  approxi- 
mately, as  check;  when  adding  a  column  of  figures,  it  should  be 
added  first  downward,  then  as  check  upward,  etc. 

Where  an  exact  calculation  is  required,  first  the  magnitude 
of  the  quantity  should  be  estimated,  if  not  already  known, 
then  an  approximate  calculation  made,  which  can  frequently 
be  done  mentally,  and  then  the  exact  calculation;  or,  inversely, 
after  the  exact  calculation,  the  result  may  be  checked  by  an 
approximate  mental  calculation. 


NUMERICAL  CALCULATIONS.  273 

Where  a  scries  of  values  is  to  be  calculated,  ii  is  advisable! 
first  to  calculate  a  few  individual  points;  and  then,  entirely 
independently,  calculate  in  tabular  form  the  series  of  values, 
and  then  use  the  previously  calculated  values  as  check.  Or. 
inversely,  after  calculating  the  series  of  values  a  few  points 
should  independently  be  calculated  as  check. 

When  a  scries  of  value-  is  calculated,  it  is  usually  easier  to 
-(•cure  reliability  than  when  calculating  a  single  value,  since 
in  the  former  case  the  different  values  check  cadi  other.  There- 
fore it  is  alwavs  advisable  to  calculate  a  number  of  values. 
that  is,  a  short  curve  branch,  even  if  only  a  single  point  is 
required.     After  calculating  ;i  series  of  values,  they  are  plotted 

as  a  curve  to  see  whether  they  give  a  .-t th  curve.      If  the 

entire  curve  i-  irregular,  the  calculation  should  be  thrown  away, 
and  the  entire  work  done  anew,  and  if  this  happens  repeatedly 
with  the  same  calculator,  the  calculator  is  advised  to  find 
another  position  more  in  agreemenl  with  his  mental  capacity. 
If  a  single  poinl  of  the  curve  appears  irregular,  this  points  to 
an  error  in  its  calculation,  and  the  calculation  of  the  point  is 
checked;  if  the  error  is  not  found,  this  point  is  calculated 
entirely  separately,  since  it  is  much  more  difficult  to  find  an 
error  which  has  been  made  than  it  is  to  avoid  making  an 
error. 

169.  Some  of  the  mosl  frequenl  numerical  errors  are: 

1.  The  decimal  error,  that  is,  a  misplaced  decimal  point. 
This  should  not  be  possible  in  the  final  result,  since  the  magni- 
tude of  the  latter  should  by  judgment  or  approximate  calcula- 
tion be  known  sufficiently  to  exclude  a  mistake  by  a  factor  in. 
However,  under  a  square  root  or  higher  root,  in  the  exponent 
of  a  decreasing  exponential  function,  etc.  a  decimal  error  may 
occur  without  affecting  the  result  so  much  as  to  he  immediately 
noticed.  The  same  is  the  case  if  the  decimal  error  occurs  in  a 
term  which  is  relatively  small  compared  with  the  other  term.-, 
and  thereby  does  not  affect  the  result  very  much.  For  instance, 
in  the  calculation  of  the  induction  motor  characteristics,  the 
quantity  /v'  +  x-x{-  appears,  and  for  small  values  of  the  slip  s, 
the  second  term  s-xi-  is  small  compared  with  r{2,  so  that  a 
decimal  error  in  it  would  affect  the  total  value  sufficiently  to 
make  it  seriously  wrong,  but  not  sufficiently  to  be  obvious. 

2.  Omission  of  the  factor  or  divisor  2. 


274  ENGINEERING  MATHEMATICS. 

3.  Error  in  the  sign,  that  is,  using  the  plus  sign  instead  of 
the  minus  sign,  and  inversely.  Here  again,  the  danger  is 
especially  great,  if  the  quantity  on  which  the  wrong  sign  is 
used  combines  with  a  larger  quantity,  and  so  does  not  affect 
the  result  sufficiently  to  become  obvious. 

4.  Omitting  entire  terms  of  smaller  magnitude,  etc. 


APPENDIX  A. 
NOTES  ON  THE  THEORY  OF  ]  UNCTIONS. 

A.  General  Functions. 

170.  The  mosl  general  algebraic  expression  of  powers  of 
x  and  //, 

F{x,y)  =  (aoo  : "<>;•''  -  "<>-'•' '-'  -  •  •  •)  +  (ai0+anx-\  ai2x2  +  .  .  .);/ 

+  (a-20+Cl2lX  +  <i.  "... 

+  (a„0+a„ix+a  r=0 I 

is  the  implicit  analytic  function.  It  relates  7  and  x  so  thai  to 
every  value  of  x  there  correspond  n  value-  of  y,  and  to  every 
value  of  y  there  correspond  m  values  of  x,  if  m  is  the  exponenl 
of  t  he  highesl  power  of  x  in    I  . 

Assuming  expression  1  solved  for  y  (which  usually  cannot 
be  carried  out  in  final  form,  as  it  requires  the  solution  of  an 
(•(luation  of  the  //th  order  in  //,  with  coefficients  which  are 
expressions  of  x),  the  explicit  analytic  function, 

y=fix) 2 

is  obtained.  Inversely,  solving  the  implicit  function  (1)  for 
x,  that  is,  from  the  explicit  function  2  .  expressing  x  as 
function  of  //.  gives  the  reverse  function  of  (2);  thai  is 

x=/  3 

In  the  general  algebraic  function,  in  its  implicit  form  I  . 
or  the  explicit  form  2  ,  or  the  reverse  function  :;  ,  x  and  y 
are  assumed  as  general  numbers;  that  i.-,  as  complex  quan- 
tities; thus, 

./•  =  .ri+/.r2;  ] 

I 

y=yi+]fj2,  J 

and  likewise  are  the  coefficients  Oqo,  "m  ...",„. 


27(3  ENGINEERING  MATHEMATICS. 

If  all  the  coefficients  a  are  real,  and  x  is  real,  the  corre- 
sponding n  values  of  y  are  either  real,  or  pairs  of  conjugate 
complex  imaginary  quantities:  y\  +jy-2  and  y\  —  jyz. 

171.  For  n=l,  the  implicit  function  (1),  solved  for  //,  gives 
the  rational  function, 

a()()+g()i.r+<7o2:r2  +  .  .  . 
aio+aiix  +  ai2x2  +  -  •    ' 

and  if  in  this  function  (5)  the  denominator  contains  no  x,  the 
integer  function, 

y  =  ao+a1x+a2x2  +  .,.+amxm,     ,     .     .     .     ((3) 

is  obtained. 

For  n,  =  2.  the  implicit  function  (1)  can  be  solved  for  y  as  a 
quadratic  equation,  and  thereby  gives 

-  (awJranx  +  aX2x'--\-...)± 

\    (a^  +  anX  +  a^2  +  ...)2  -4(am  +  alnX  +  a0.^2  +  ...)(a,n  +  UnX  +  a2rX2 ^-  ...) 
y~  2{a20  +  a21x  +  a2.^2  +  ...)  '  V> 

that  is,  the  explicit  form  (2)  of  equation  (1)  contains  in  this 
case  a  souare  root. 

X 

For  n>2,  the  explicit  form  y=f(x)  cither  becomes  very 
complicated,  for  n  =  3  and  n  =  4,  or  cannot  be  produced  in 
finite  form,  as  it  requires  the  solution  of  an  equation  of  more 
than  the  fourth  order.  Nevertheless,  y  is  still  a  function  of 
x,  and  can  as  such  be  calculated  by  approximation,  etc. 

To  find  the  value  y\,  which  by  function  (1)  corresponds  to 
x  =  xi,  Taylor's  theorem  offers  a  rapid  approximation.  Sub- 
stituting X]  in  function  (1)  gives  an  expression  which  is  of 
the  nth  order  in  y,  thus:  F(x\y),  and  the  problem  now  is  to 
find  a  value  y\,  which  makes  F(x\,y\)=§. 

However, 

v     -nr       n     7  dF(xi,y)    h2d?F(xuy) 
F(xhyi)  =  F(xhy)+h — ^-+12       ^2 J  +...,  •    (8) 

where  h  =  yx  —  y  is  the  difference  between  the  correct  value  y\ 
and  any  chosen  value  //. 


APPENDIX  A. 


277 


Neglecting  the  higher  orders  of  the  small  quantity  h,  in 
(8),  and  considering  that  F(xi,yi)  =  0,  gives 


h 


F(xh  y) 

dF(xuy)' 

dy 


(9) 


and  herefrom  is  obtained  yi  =  y+h,  as  first  approximation. 
Using  this  value  of  y\  as  y  in  (9)  gives  a  second  approximation, 
which  usually  is  sufficiently  close. 

172.  New  functions  are  defined  by  the  integrals  of  the 
analytic  functions  (1)  or  (2),  and  by  their  reverse  functions. 
They  are  called  Abelian  integrals  and  Abelian  functions. 

Thus  in  the  most  general  case  (1),  the  explicit  function 
corresponding  to  (1)  being 


the  integral, 


z=Cf(x)dx, 


(2) 


then  is  the  general  Abelian  integral,  and  its  reverse  function, 

x=<f>(z), 

the  general  Abelian  function. 

(a)  In  the  case,  n=l,  function  (2)  gives  the  rational  function 
(5),  and  its  special  case,  the  integer  function  (6). 

Function  (0)  can  be  integrated  by  powers  of  x.  (5)  can  be 
resolved  into  partial  fractions,  and  thereby  leads  to  integrals 
of  the  following  forms : 

(1)      Cxmdx) 


dx 


■a 

dx 


(3)     J  \x-a)> 
s       C  dx 


(10) 


278  ENGINEERING  MATHEMATICS. 

Integrals  (10),  (1),  and  (3)  integrated  give  rational  functions, 
(10),  (2)  gives  the  logarithmic  function  log  (x— a),  and  (10),  (4) 
I  he  arc  function  arc  tan  .r. 

As  the  arc  functions  are  logarithmic  functions  with  complex 
imaginary  argument,  this  case  of  the  integral  of  the  rational 
function  thus  leads  to  the  logarithmic  function,  or  the  loga- 
rithmic integral,  which  in  its  simplest  form  is 


/ 


dx     ,  ,     v 

j  =  logx, (11) 


and  gives  as  its  reverse  function  the  exponential  f nudum, 

x=sz (12) 

It  is  expressed  by  the  infinite  series, 

z2        -,3        ?4 

sz  =  l+.e+n7+^+fr  + (13) 

as  seen  in  Chapter  II,  paragraph  53. 

173.  b.  In  the  case,  n  =  2,  function  (2)  appears  as  the  expres- 
sion (7),  which  contains  a  square  root  of  some  power  of  x.  Its 
first  part  is  a  rational  function,  and  as  such  has  already  been 
discussed  in  a.     There  remains  thus  the  integral  function, 


rx%+blx  +  b2x2  +  .  .  .+bl,x') 


z=  U!T      1      ,'••       '-dx.   .     .     .     (14) 

Co  +  ClX+C-2-X2  +  .  ■  ■ 


This  expression  (14)  leads  to  a  series  of  important  functions. 
(1)  Forp  =  l  or  2, 

C   Vbo  +  bxx  +  box'2 
J  cn+CiX+c-2X-+.  .  • 

By  substitution,  resolution  into  partial  fractions,  and 
separation  of  rational  functions,  this  integral  (11)  can  be 
reduced  to  the  standard  form, 

f     dx 
z=  I  — (10) 

In  the  case  of  the  minus  sign,  this  gives 

/dx 
-7=  ===arc  sin  x, (17) 
\  I  —  x2 


APPENDIX   A. 


279 


and  as  reverse  functions  thereof,  there  arc  obtained   the  trigo- 
no i itctric  functions. 

.r  =  sin  z,  1 

[ (18) 

Vl  —  X2  =  cos  z.  j 

In  the  ease  of  the  plus  sign,  integral  (16)  gives 


z  = 


/('  r 
=  =- log{  \  1  +x2  —  x }  =  are  sinh  x,     .    (19) 
v    .1    r  «£ 


and  reverse  functions  thereof  are  the  hyperbolic  functions, 

-  +  z ~— z 


Vl+X2  = 


2 

£  +  z  +  3' 


=sinh  z\ 
cosh.?. 


(20) 


The  trigonometric  functions  are  expressed  by  the  series; 


23     z5     z1 

sin  z  =  z—  ut+7- —  -p-  + 
\6      o      \i 


yl  y\  gQ 

COS  2=1  —  FT +T7  —  \7^  +  - 

2       4      () 


•  • 


•     (2D 


as  seen  in  Chapter  II,  paragraph  58. 

The  hyperbolic  functions,  by  substituting  for  e+z  and  e~* 
the  series  (13),  can  be  expressed  by  the  series: 


z3     z5     z7 


sinh  2  =  2+777+7-^+7^  +•  •  •  I 
3     b      / 


Z~       Z^        ^ 

COsh2=l+7-r+fr+J7r+.  . 

2       4       () 


(22) 


174.  In  the  next  case,  p=3  or  4. 


\  b0  +  &i£  +  ?>2-r2  +  6:5.r3  +  6  4x4 


Cu+CiX+C2X^  +  .  . 


dx, 


(23) 


already  leads  beyond  the  elementary  functions,  that  is,   (23) 
cannot  be  integrated  by  rational,  logarithmic  or  arc  functions, 


280 


EXdLXEERlNG  MATH  EM  AT  1<  'S. 


but  gives  a  new  class  of  functions,  the  elliptic  integrals,  and 
their  reverse  functions,  the  elliptic  functions,  so  called,  because 
they  bear  to  the  ellipse  a  relation  similar  to  that,  which  the 
trigonometric  functions  bear  to  the  circle  and  the  hyperbolic 
functions  to  the  equilateral  h}rperbola. 

The  integral  (23)  can  be  resolved  into  elementary  functions, 
and  the  three  classes  of  elliptic  integrals: 


a 


Mi 


U-2 


J 


dx 


Vx(i—x)(i—c2xy 

xdx 
Vx{l  -  x)  (I  -  c2x) ' 
dx 


(x-b)Vx(l-x){l-c-x)    J 


(24) 


(These  three  classeb  of  integrals  may  be  expressed  in  several 
different  forms.) 

The  reverse  functions  of  the  elliptic  integrals  arc  given  by 
the  elliptic  functions : 

Vz=sin  am(u,  c)\ 


1  —  x  =  cos  am(u,  c);   ■ 


(25) 


VI  —c2x  =  Jam(u,  c); 


known,  respectively,  as  sine-amplitude,  cosine-amplitude,  delta- 
amplitude. 

Elliptic  functions  are  in  some  respects  similar  to  trigo- 
nometric functions,  as  is  seen,  but  they  are  more  general, 
depending,  as  they  do,  not  only  on  the  variable  x,  but  also  on 
the  constant  c.  They  have  the  interesting  property  of  being 
doubly  periodic.  The  trigonometric  functions  arc  periodic,  with 
the  periodicity  2tt,  that  is,  repeat  the  same  values  after  every 
change  of  the  angle  by  2n.  The  elliptic  functions  have  two 
periods  pi  and  p-2,  that  is, 

sin  am(u  +  npi  Jrmp2,  c)  =sin  am(u,  c),  etc.;      .     (26) 

hence,   increasing  the   variable   u  by  any   multiple   of  either 
period  pi  and  />2,  repeats  the  same  values. 


APPENDIX  A. 


281 


The  two  periods  arc  given  by  the  equations, 

dx 


Pi 


£ 


2\  x{l-x)(l-c2x)' 


P~  ~X     2 \  .r(.l-x)(l-c2.r) ' 


(27) 


175.  Elliptic  functions  can  be  expressed  as  ratios  of  two 
infinite  scries,  and  these  series,  which  form  the  numerator  ami 
the  denominator  of  the  elliptic  function,  are  called  theta  func- 
tions and  expressed  by  the  symbol  0,  thus 


1        \2pi 

sin  am[u,  c)  =—-=. — -1— 
vc  -  /  ~u 


II 


i)l ; 


cos  am(u,  c)  =Jl 


2pi 


\2p\ 


Jamid,  c) 


a/F 


l-c2        mi 

\ip\ 


\2pi 

,  ~!< 

•  Is— 

2pi 


■cr- 


(28) 


and  the  four  0  functions  may  be  expressed  by  the  series: 
0o(x)  =  1  —2q  cos  2x+2q4  cos  4.r  —  2q°  cos  Qx  -I — .  .  .  ; 

di(x)  =2g1/4  sin  .r  -2</9/4  sin  3z  +  27T  sin  5x  -  +  . .  .  : 

25 

02(x)  =2ql/4  cos  x+2^9/4  cos  3x+2q*  cos  5x  +  .  .  .  : 
d3(x)  =  1  +2ry  cos  2x  +2q*  cos  4.r  +27°  cos  (>.r  +  .  .  .  , 


(29) 


\  here 


1  •     P2 

o=£a     and     a=?7r — . 

*    pi 


(30) 


In  the  case  of  integral  function   (14),  where  p>4,  similar 
integrals   and   their  reverse   functions   appear,    more   complex 


282  ENGINEERING  MATHEMATICS. 

than  the  elliptic  functions,  and  of  a  greater  number  of  periodici- 
ties.    They  are  called   hyperelliptic  integrals  and  hyperelliptic 

functions,  and  the  latter  are  again  expressed  by  means  of  auxil- 
iary functions,  the  hyperelliptic  0  functions. 

176.  Many  problems  of  physics  and  of  engineering  lead  to 
elliptic  functions,  and  these  functions  thus  are  of  considerable 
importance.  For  instance,  the  motion  of  the  pendulum  is 
expressed  by  elliptic  functions  of  time,  and  its  period  thereby 
is  a  function  of  the  amplitude,  increasing  with  increasing  ampli- 
tude: that  is,  in  the  so-called  "  second  pendulum,"  the  time  of 
one  swing  is  not  constant  and  equal  to  one  second,  but  only 
approximately  so.  This  approximation  is  very  close,  as  long 
as  the  amplitude  of  the  swing  is  very  small  and  constant,  but 
if  the  amplitude  of  the  swing  of  the  pendulum  varies  and 
reaches  large  values,  the  time  of  the  swing,  or  the  period  or 
the  pendulum,  can  no  longer  be  assumed  as  constant  and  an 
exact  calculation  of  the  motion  of  the  pendulum  by  elliptic 
functions  becomes  necessary. 

In  electrical  engineering,  one  has  frequently  to  deal  with 
oscillations  similar  to  those  of  the  pendulum,  for  instance, 
in  the  hunting  or  surging  of  synchronous  machines.  In 
general,  the  frequency  of  oscillation  is  assumed  as  constant, 
but  where,  as  in  cumulative  hunting  of  synchronous  machines, 
the  amplitude  of  the  swing  reaches  large  values,  an  appreciable 
change  of  the  period  must  be  expected,  and  where  the  hunting 
is  a  resonance  effect  with  some  other  periodic  motion,  as  the 
engine  rotation,  the  change  of  frequency  with  increase  of 
amplitude  of  the  oscillation  breaks  the  complete  resonance  and 
thereby  tends  to  limit  the  amplitude  of  the  swing. 

177.  As  example  of  the  application  of  elliptic  integrals,  may 
be  considered  the  determination  of  the  length  of  the  arc  of  an 
ellipse. 

Let  the  ellipse  of  equation 

P+&"1 (31) 

be  represented  in  Fig.  93,  with  the  circumscribed  circle, 

x2+y2  =  a2 (32) 


APPENDIX  A. 


283 


To  every  point  P  =  x,  y  of  the  ellipse  then  corresponds  a 
point  P\  =  x,  )/i  on  the  circle,  which  lias  the  same  abscissa  x, 
and  an  angle  Q  =  AOP\. 

The  air  of  the  ellipse,  from  A  to  P,  then  is  given  by  the 
integral, 

L  =  a      —  — (33) 

J02\  z{l-z){l-cH) 

where 


xy  \V-fc* 

2=sin^  0=  \  —  J       and     c=^ 


a 


is  the  eccentricity  of  the  ellipse. 


.     (34) 


Fig.  93.     Rectification  of  Ellipse. 

Thus  the  problem  leads  to  an  elliptic  integral  of  the  first 
and  of  the  second  class. 

For  more  complete;  discussion  of  the  elliptic  integrals  and 
the  elliptic  functions,  reference  must  be  made  to  the  text-books 
of  mathematics. 


B.  Special  Functions. 

178.  Numerous  special  functions  have  been  derived  by  the 
exigencies  of  mathematical  problems,  mainly  of  astronomy,  but 
in  the  latter  decades  also  of  physics  and  of  engineering.  Some 
of  them  have  already  been  discussed  as  special  cases  of  the 
general  Abelian  integral  and  its  reverse  function,  as  the  expo- 
nential, trigonometric,  hyperbolic,  etc.,  functions. 


28 1  ENGINEERING  MA  Til  EM  A  TI(  'S. 

Functions  may  be  represented  by  an  infinite  series  of  terms; 
that  is,  as  a  -sum  of  an  infinite  number  of  terms,  which  pro- 
gressively decrease,  that  is,  approach  zero.  The  denotation  of 
the  terms  is  commonly  represented  by  the  summation  sign  2. 

Thus  the  exponential  functions  may  be  written,  when 
defining, 

[0  =  1;         |/?  =  1X2X3X4X.  .  .Xn, 

as 

x2     r3  t»    r" 

ex  =  i+x+     +     +m  m  .  =  S»7— ,     ....     (35) 
|2     |3  0    \n 

xn> 
which  means,  that  terms  , —  are  to  be  added  for  all  values  of  n 
»  n 

from  n  =  0  to  n  =  oc  . 

The  trigonometric  and  hyperbolic  functions  may  be  written 
in  the  form : 


sin  x =x  - 

X3      X5      X7 
X2      T4      X6 

■|2+l4-j6+- 

.r3    x5    x? 

1       r 

X2      J"4      x6 

■x                r2> 

vn  (       IV" 

+  1 

.     (36) 

o                2n 

+1 ' 

COS  X  =  1  - 

inh  x-=x-\ 

oo                 f2n 
i)               |2ft 

rjO          v.2«  +1 

• 

•     (37) 
.     (38) 

"     o    2ft  +  l  ••     ' 

£    X2n 

cosh  x  =  l  +nr+!T+TT+.  •  •  =  Snn-. 
|-      |4      |6  o    |2ft 


(39) 


Functions  also  may  be  expressed  by  a  series  of  factors; 
that  is,  as  a  product  of  an  infinite  series  of  factors,  which  pro- 
gressively approach  unity.  The  product  series'  is  commonly 
represented  by  the  symbol  ]T. 

Thus,  for  instance,  the  sine  function  can  be  expressed  in  the 
form, 


**-<^-m-&----W-£i- 


(40) 


179.  Integration  of  known  functions  frequently  leads  to  new 
functions.     Thus   from    the   gancral   algebraic    functions   were 


APPENDIX  A. 


285 


derived  the  Abelian  functions.     In  physics  and  in  engineering, 

integration  of  special  functions  in  this  manner  frequently  leads 

to  new  special  functions. 

For  instance,  in  the  study  of  the  propagation  through  space, 

of  the  magnetic  field  of  a  conductor,  in  wireless  telegraphy, 

lightning  protection,  etc.,  we  get  new  functions.     If  i=f  (t) 

is  the  current  in  the  conductor,  as  function  of  the  time  /,  at  a 

distance  x  from  the  conductor  the  magnetic  field  lags  by  the 

x 
time  /i  =  ^,  where  S  is  the  speed  of  propagation  (velocity  of 

light).     Since   the   field   intensity   decreases   inversely   propor- 
tional to  the  distance  x,  it  thus  is  proportional  to 


A'-s 


y=- 


(41) 


and  the  total  magnetic  flux  then  is 


"t^1 


■/ 


z=  I  ydx 


■S 


/c-i 


X 


dx. 


m 


Tf  the  current  is  an  alternating  current,  that  is,  f  (t)  a 
trigonometric  function  of  time,  equation  (42)  leads  to  the 
functions, 


u=      — 


I  — —  dx; 

/COS  X    , 
ax. 
x 


(43) 


If  the   current   is  a  direct   current,   rising  as  exponential 
function  of  the  time,  equation  (42)  leads  to  the  function, 


IV 


/'  exdx 
x~- 


(44) 


286 


ENGINEERING  MA  THEM.  1  TICS. 


Substituting  in   (43)   and    (44),   for  sin  x,  cos  x,    sx  their 
infinite  scries   (21)  and   (13),  and  then   integrating,  gives  the 


following: 


sin  x 


x3      .r5       x7 
x  313     5|5     717 


.     (45) 


feos.r         ,  x2       x4       X* 

I    \r-'/-r  =  1°-J-2i:>+4l4"^+~ 

*-^  I J ,  I 

I  —dx  =  \os,x+x+-^:  +^r-  +.  .  . 
/  .'•  &  22     33 

For  further  discussion  and  tables  of  these  functions  see 
'•'  Theory  and  Calculation  of  Transient  Electric  Phenomena  and 
Oscillations,"  Section  III,  Chapter  VIII,  and  Appendix. 

.80.  If  „-/(*)  is  a  function-  of  x,  and  ,-f /&,-«,) 

its  integral,  the  definite  integral,  Z  =  I  f(x)dx,   is    no  longer 
a  function  of  x  but  a  constant, 

For  instance,  if  y -=  c{x — ti)2 ,  then 

c(x  —  n)3 


and  the  definite  integral  is 


3 


■£« 


x — n)2dx  =  -\(b  —  ?i)3—  (a  —  n)3} . 


This  definite  integral  does  not  contain  .r,  but  it  contains 
all  the  constants  of  the  function /(.r),  thus  is  a  function  of 
these  constants  c  and  n,  as  it  varies  with  a  variation  of  these 
constants. 

In  this  manner  new  functions  may  be  derived  by  definite 
integrals. 

Thus,  if 

y=f{x,v,v...) (46) 

is  a  function  of  x,  containing  the  constants  u,  v  .  .  . 


APPENDIX  A.  2S7 

The  definite  integral, 

Z  =■-  J  f{x,  u,  v...  )dx, (47) 

is  not  a  function  of  x,  but  still  is  a  function  of  u,  v  .  .  .  ,  and 
may  be  a  new  function. 
181.  For  instance,  let 

y=s-xxu~1; (48) 

then  the  integral, 


£-xxu-ldX} 


(49) 


is  a  new  function  of  u,  called  the  gamma  function. 

Some  properties  of  this  function  may  be  derived  by  partial 
integration,  thus: 

r(u+l)=ur(u); (50) 

if  n  is  an  integer  number, 

r(u)  =  (u-l){u-2)...(u-n)r{u-n),    .    .     (51) 
and  since 

m)=\, (52) 

if  u  is  an  integer  number,  then, 

r{u)  =  \u-l.         (53) 

C.    Exponential,  Trigonometric  and  Hyperbolic  Functions. 

(a)  Functions  of  Real  Variables. 

182.  The  exponential,  trigonometric,  and  hyperbolic  func- 
tions are  defined  as  the  reverse  functions  of  the  integrals, 


fdx    . 
I  —  =  l°g 


a.  u=    I—  =logx. (54) 

and:  x=s« (55) 

C     dx 

0.  u=  I     /  =arc  sin  x;     .......     (5G) 

J  vl-x2 


288  ENGINEERING  MATHEMATICS. 

and:  x=sinw, (57) 

Vl-x2  =  cosu, (58) 

/dx 
■^===-\og{VT+^-x};.    .     .    .     (59) 


c.  u  = 


.'I  . c~" 


and  x=- — - — = sinh  it;      ....     (60) 

Vl+.r2  = ~ — =coshw (61) 

From  (57)  and  (58)  it  follows  that 

sin2  u  +cos2?t  =  l (62) 

From  (60)  and  (61)  it  follows  that 

cos2/m-sin2/m=l (63) 

Substituting  (—x)  for  x  in  (56),  gives  (— ?/)  instead  of  it, 
and  therefrom, 

sin  (— u)  =  —  sin  u (64) 

Substituting;   (— u)   for   u   in   (60),   reverses  the  sign   of    x, 
that  is, 

sinh  (— u)  =  —  sinh  it.         ....     (i'u)) 

Substituting  (—x)  for  x  in  (58)  and  (61),* does  not  change 
the  value  of  the  square  root,  that  is, 

cos  {  —  u)=  cos  u, (66) 

cosh  (—it)  =  cosh  u,       (07) 

Which  signifies  that  cos  u  and  cosh  u  are  even  functions,  while 
sin  :<  and  sinh  u  are  odd  functions. 

Adding  and  subtracting  (60)  and  (61  ),  gives 

e±«=coshw±sinh« (68) 


APPENDIX  A.  289 

(h)  Functions  of  Imaginary  Variables. 
183.  Substituting,  in  (56)  and  (59),  x=  —jy,  thus  y  =  jx,  gives 

C     dx.  C     dx 

(56.)      ,-J^=;  (50.)      «=J^=; 


x  =  sin  ;/ ;  a; = sinh  w  = 


9        > 


Vl  +  j-2  =  cos  u ;  Vl  +  x-  =  cosh  w  =  ■ 


0       ' 


hence,  p  =  (  -===,  hence,  71* =  (  — = 

J  vl+v2  J  vl 


iy 


2> 


!f  J  vl-// 

£ju_  £—  7« 

?/= sinh  jit  = ^ ;  ?/  =  sin///;     .     .     .     (69) 


r?»  -f   c-JW 


Vl+2/2  =  cosh/u= ~ ;      Vl—y2  =  cos  ju\    .     .     .     (70) 

Resubstituting  x  in  both 

sinh/';/      gu—g-iw  t  e"  —  £-«     sin  m 

£=sinw= r^— =  — — : ■;   x=$mhu  =  — ■=. —  =  — ~ ;  {/I) 

1  -'  2  ./ 


£"  + 


«    _|_     c— " 


Vl^-  x2  =  cos  ?/  =  cosh  ju  vT+  x2  =  cosh  u=- 

'2 

=  1-ryL — 5  =  cos/u.     .     (72) 


Adding  and  subtracting, 

s±3"  =  cos  u±j  sin  w=cosh  /?/±sinh  ju 
and  £±u  =  cosh  u  ±sinh  w=cos  juT  j  sin  ju.     .     .     (73) 

(c)  Functions  of  Complex  Variables 

184.  It  is: 

su±jv=  £«£±*»=£«(cos  v±/sin  v);      .     .     .     (74) 


290 


ENGINEERING  MA  THEM  A  Til 'S. 


sin  (it±/iO=sin  u  cos  p±cos  u  sin  p 

£''  -he-*  . 
=  sin  u  cosh  v  ±  j  cos  m  sinh  ?;  =  — - — sm  w  ±  ] — -} — cos  u ; 


.£v-£-v 


}■     (75) 


cos(w  ±  ji')  =  cos  u  cos  ji'  =F  sin  u  sin  jv 

=  cos  u  cosh  vT  ]  sin  «  sinh  v  =  — ^ — cos  w=F  J — « —  sin  w; 


(76) 


sinh  (it  ±p) 


rK±JD g—uTjV  £'<  j—  U 


.£"+£ 


x     I    »  —  a 


— — cosv±j — ~y 


-smi> 


■     (77) 


=  sinh  u  cos  v±j  cosh  it  sin  v ; 


.£"-£"" 


£!4±Jt'  _J_£—  UTJC  £U  -\- £~ " 

cosh(u±p)  =  -      -s-  -« — cosi>±j — q — sinr' 


>     (78) 


=  cosh  u  cos  v ±j  sinh  w  sin  v; 


etc. 


(d)  Relations. 

185.  From  the  preceding  equations  it  thus  follows  that  the 
three  functions,  exponential,  trigonometric,  and  hyperbolic, 
are  so  related  to  each  other,  that  any  one  of  them  can  be 
expressed  by  any  other  one,  so-  that  when  allowing  imaginary 
and  complex  imaginary  variables,  one  function  is  sufficient. 
As  such,  naturally,  the  exponential  function  would  generally 
be  chosen. 

Furthermore,  it  follows,  that  all  functions  with  imaginary 
and  complex  imaginary  variables  can  be  reduced  to  functions 
of  real  variables  by  the  use  of  only  two  of  the  three  classes 
of  functions.  In  this  case,  the  exponential  and  the  trigono- 
metric functions  would  usually  be  chosen. 

Since  functions  with  complex  imaginary  variables  can  bo 
resolved  into  functions  with  real  variables,  for  their  calculation 
tables  of  the  functions  of  real  variables  are  sufficient. 

The  relations,  by  which  any  function  can  be  expressed  by 
any  other,  are  calculated  from  the  preceding  paragraph,  by 
the  following  equations: 


APPENDIX  A. 


291 


e±tt=cosh  ii±sinh  u=cos  ju^fjsin  ju; 

£±jv  =  cog  y  _j_  y  gjn  v  _  co^j1  p  _j_  y  ^ jn}1  yy . 

£tti,B  =  cM  (cos  i>±  j  sin  v), 

sinh  /'?/     c;'"—  e-'" 
sin  u  = r^— = —. ; 

J  2J 

cV ~—v 

sinp=jsmh  v  =  j-—^—', 
sin  (it±;?;)=sin  u  cosh  r±/ cos  u  sinh  r 


£»  +  £-» 


C-D -—  '-' 


-: — Sill  U  ±J 


C<  )S  U 


cos  u  =  cosh  ju 


cos  jv  =  cosh  y 


"        2~       : 

£J»  _j_  £—  jv 


cos  (u±jv)  =cos  w  cosh  u^?  sin  '/sinh 


fV—  ff— v 


ct> .— u 


— ~ —  cos  itlj — s sili  u 


sinh  u  =  ■ 


CU -—  U 


Sill    _/« 


£>V—    £—JV 

sinh  p  =  /  sin  v= -1 — ; 


sinh  (u±jv)  =sinh  u  cos  w  ±  /  cosh  u  sin  u 

= F> COS  V±y — —  sili  ?:; 


cosh  ii 

cosh  /y  =  cos  i' 


=  cosjw; 

■jv  _|_  £  —  jv 

2 


cosh  (u±jv)  =  cosh  w  cos  v  ±/  sinh  u  sin  w 

= — 2  —  cos  v±j- — 2" —  sin  v- 


292  ENGINEERING  MA  THEM  A  TICS. 

And  from  (b)  and  (d),  respectively  (c)  and  (e),  it  follows  that 

sinh  (u  +  jv)  =  j sin  (±v—  ju)  =  ±j sin  (v ± ju) ;  1 
cosh  (it  ±  jv)  =  cos  (*>  =F  ju) .  J 

Tables  of  the  exponential  functions  and  their  logarithms, 
and  of  the  hyperbolic  functions  with  real  variables,  are  given 
in  the  following  Appendix  B. 


APPENDIX   B. 

TWO    TABLES    OF    EXPONENTIAL    AND    HYPERBOLIC 

FUNCTIONS. 

Table  I. 


£-2.7183, 


log  £  =  0.4343. 


X 

XIO-3 

X10-2 

xio-i 

XI 

1.0 

0.999 

0.990 

0.905 

0.368 

1.2 
1.4 
1.6 
1.8 

0.988 
0.986 
0.984 
0.982 

0 .  887 
0.869 
O.K.-,_> 
0.835 

0.301 
0.247 
1 1  202 
0.165 

2.0 

0.998 

0.980 

0.819 

0.135 

2.5 
3.0 
3.5 
4.0 
4.5 

0.997 
0.996 

0  975 
0.970 
0.966 
0.961 
0.956 

0-779 
0.741 

0.70.". 
0.670 
0.638 

0.082 
0.050 
0.030 

0.018 
0.011 

5.0 

0.995 

0.951 

0.607 

0.007 

6 

7 
8 
9 

0 .  994 
0.993 
0.992 
0.991 

0.942 
0.932 
0.923 
0.914 

0 .  549 
0.497 

0.449 

0.407 

0.002 
0.001 
0.000 

10 

0.990 

0.905 

0.368 

293 


294 


ENGINEERING  MA T HEMATICS. 


Table  II. 


EXPONENTIAL  AND   HYPERBOLIC   FUNCTIONS. 

£  =  2.718282~2.7183,  log  £  =  0.4342945-0.4343. 

coshx  =  h{e  +  x  +  £-x\,  smhx  =  h\e  +  x-e-x\. 


p.p. 

434 

435 

0 

0 

0 

0.1 

43 

43 

0.2 

87 

87 

0.3 

130 

130 

0.4 

174 

174 

0.5 

217 

217 

0.6 

261 

261 

0.7 

304 

304 

0.8 

347 

348 

0.9 

391 

391 

1.0 

434 

435 

Jlog 

X 

log£  +  Z 

E±x 

log  e  x 

e  +  x 

£-* 

cosh  x 

sinh  x 

x 

0 

0 

434 
435 
434 
434 

434 

435 

434 
434 
435 

434 

0 

1 

1 

1 

0 

0 

0.001 
0.002 

0.000434 
0.000869 

9.999566 
9.999131 

1.00100 
1.00200 

0.99900 
0.99800 

1.00000 
1 . 00000 

0.00100 
0 . 00200 

J.  001 
0.002 

0.003 

0.001303 

9.998697 

1.00301 

0.99700 

1.00000 

0.00300 

0.003 

0.004 

0.001737 

9.998263 

1.00401 

0.99601 

1.00001 

0.00400 

0.004 

0.005 

0.002171 

9.997829 

1.00501 

0.99501 

1.00001 

0 . 00500 

0.005 

0.006 

0.002606 

9.997394 

1.00602 

0.99402 

1.00002 

0.00600 

0.006 

0.007 

0.003040 

9.996960 

1.00702 

0.99302 

1 . 00002 

0.00700 

0.007 

0.008 

0.003474 

9.996526 

1.00803 

0.99203 

1.00003 

0.00800 

0.008 

0.009 

0.003909 

9.996091 

1.00904 

0.99104 

1.00004 

0 . 00900 

0.009 

0.010 

0.004343 

9 . 995657 

1.01005 

0 . 99005 

1.00005 

0.01000 

0.010 

0.012 

0.005212 

9 . 994788 

1.01207 

0.98807 

1.00007 

0.01200 

0.012 

0.014 

0.006080 

9.993920 

1.01410 

0.98610 

1.000 10 

0.01400 

0.014 

0.016 

0.006949 

9.993051 

1.01613 

0.98413 

1.00013 

0.01600 

0.016 

0.018 

0.007817 

9.992183 

1.01816 

0.98216 

1.00016 

0.01800 

0.018 

0.020 

0.008686 

9.991314 

1.02020 

0.98020 

1.00020 

0.02000 

0.020 

0.025 

0.010857 

9.989143 

1.02531 

0.97531 

1.00031 

0 . 02500 

0.025 

0.030 

0.013029 

9.986971 

1.03046 

0.97045 

1.00046 

0.03000 

0.030 

0.035 

0.015200 

9.984800 

1.03562 

0.96561 

1.00062 

0.03500 

0.035 

0.040 

0.017372 

9.982628 

1.04081 

0.96079 

1 . 00080 

0.04001 

0.040 

0.045 

0.019543 

9 . 980457 

1 . 04603 

0.95600 

1.00102 

0.04502 

0.045 

0.050 

0.021715 

9.978285 

1.05127 

0.95123 

1.00125 

0 . 05003 

1 

0.050 

0.06 

0 . 026058 

9.973942 

1.06184 

0.94176 

1.001800.06004 

0.06 

0.07 

0.030401 

9.969599 

1.07251 

0.93239 

1. 00245 !  0.07006 

0.07 

0.08 

0.034744 

9 . 965256 

1.08329 

0.92312 

1.00321 ^.08008 

0.08 

0.09 

0.039086 

9.960914 

1.09417 

0.91393 

1.00405  0.09011 

0.09 

0.10 

0.043429 

9.956571 

1.10516 

0.90484 

1.00500  0.10016 

0.10 

0.12 

0.052115 

9.947885 

1.12750 

0 . 88692 

1.00721J0.12028 

0.12 

0.14 

0.060801 

9.939199 

1 . 15027 

0 . 86936 

1.00982  0.  !  1046 

0 .  14 

0.16 

0 . 069487 

9.930513 

1.17351 

0.85214 

1.01283  0.16069 

0.16 

0.18 

0.078173 

9.921827 

1.19721 

0 . 83527 

1 .01624 

0.18097 

0.18 

0.20 

0.086859 

9.913141 

1  .22140 

0.81873 

1.02006 

0.20134 

0.20 

e  +  000i  =  1.001000494, 


£-0001  =  0.99900049. 


APPENDIX  B. 


295 


Table  II — Continued. 


EXPONENTIAL   AND    HYPERBOLIC   FUNCTIONS. 


X 

log  £+* 

log  e~x 

e  +  x 

£-* 

cosh  x 

sinh  .r 

X 

0 .  20 

0.086859 

9.913141 

1.22140 

0.81873 

1.02006 

0.20131 

0.20 

0 .  25 

0.30 
0.35 
0.40 
0.45 

0.108574 
0.130288 
0.152003 
0.173718 
0.195433 

9.891  l-'ti 
9 . 8697 1 2 
9.847997 
9 . 826282 
9.804567 

1 .28403 
1 . 34986 
1.41907 

1.49183 
1.56831 

0 . 77880 
0.74082 
0 . 70469 
0.67032 
0 . 63763 

1.031  12 
1.04534 
1.06188 
1.08108 

1.10207 

0.25261 
0 . 30457 
0.35719 
0.41076 
0.46534 

0.25 
0.30 
0.35 
0.40 
0.45 

0.50 

0.217147 

9 . 782853 

1.64870 

0 . 60653 

1.12761 

0.52H  is 

0.50 

0.6 
0.7 
0.8 
0.-9 

0.260577 
0.304006 
0.347436 
0.390865 

9.739423 
9 . 695994 
9.652564 
9.609135 

1.82212 

2.0137.-) 
2.22554 
2 . 45960 

0.54881 
0.49659 
0.44933 
0.40657 

1.19546 
1.25517 
1.33744 
1.43309 

0 . 63666 
0 . 75858 
0.88811 
1.02657 

0.6 

0.7 
0.8 
0.9 

1.0 

0 . 434294 

9.565706 

2.71828 

0.36788 

1 . 54308 

1.17520 

1.0 

1.2 
1.4 
1.6 
1.8 

0.521153 
0.608012 
0.694871 
0.781730 

9.478847 
9.391988 
9.305129 
9.218270 

3.32011 
4.05520 
4.95304 
6 . 04965 

0.30119 
0.24660 

0.20190 
0.16530 

1.81065 
2.15090 

2.5771T. 
3.10715 

1 . 50946 
1.90430 
2 . 37557 
3. 14218 

1.2 
14 
1.6 

1.8 

2.0 

0.868589 

9.131411 

7 . 38906 

0.13534 

3.76220 

3 . 62686 

2.0 

2.5 
3.0 
3.5 
4.0 
4.5 

1.085736 
1.302883 
1 . 520030 
1.737178 
1.954325 

8.914264 
8 . 694 1 1 7 
8.479970 
8 . 262822 
8.045675 

12.1825 
20.0855 
33.1154 
54.5983 
90.0170 

0.082085 
0 . 049797 
0.030197 
0.018316 
0.011109 

6.1323 

10.0077 

16.57:8 

27.30X3 
45.0141 

6.0002 
10.0178 
1 6  5426 
27  2900 
'. -1)030 

2.5 
3.0 

3.5 
4.0 
4.5 

5.0 

2.171472 

7.828528 

148.413 

0.006738 

74.210 

74.203 

5.0 

6 

7 
8 
9 

2 . 605767 
3.040061 
3.474356 
3.908650 

7 . 394233 
6.959939 
6 . 525644 
6.091350 

403 . 428 
1096.63 
2980.96 
8103.08 

0.002479 
0.000012 
0.000335 
0.000123 

201.715 

201.713 

6 
7 
8 
9 

for  .r>  6 

10 

4.342945 

5 . 657055 

22026.5 

0.0000454 

10 

12 
14 
16 

18 

5.211534 
6.080123 
6.948712 
7.817301 

4 . 788466 
3.919877 
3.051288 
2 . 182699 

162755 

1202610 

8886120 

65660000 

0.0000061 
0 . 00000083 
0.00000011 
0.00000002 

12 
14 
16 

18 

20 

8 . 685890 

1.314110 

485166000 

0.00000000 

20 

INDEX. 


Abelian  integrals  and  functions,  286. 
Absolute  number,  4. 

value  of  fractional  expression,  49. 
of  general  number,  30. 
Accuracy,  loss  of,  264. 

of  approximation  estimated,  200. 
of  transmission  line  equations,  208. 
of  calculation,  262. 
of  curve  equation,  210. 
Addition,  1. 

of  general  number,  28. 
and  subtraction   of   trigonometric 
functions    102. 
Algebra  of  general  number  or  com- 
plex quantity,  25. 
Algebraic  e  pression,  27."). 

function,    75. 
Alternating  current  and  voltage  vec- 
tor, 41. 
functions,  117,  125. 
waves,  117,  125. 
Alternations,  117. 
Alternator     short     circuit      current, 

approximated,  195. 
Analytical    calculation    of    extrema, 
152. 
function,  275. 
Angle,  see  also  Phase  angle. 
Approximation  calculation,  263. 
Approximations    giving    (l+s)    and 
(l-s),201. 
of  infinite  series,  53. 
methods  of,  187. 
Arbitrary  constants  of  series,  69,  79. 
Area  of  triangle,  106. 
Arrangement   of   numerical   calcula- 
tions, 258. 
Attack,  method  of,  258. 


B 

Base  of  logarithm,  21. 

Binomial  scries  with  small  quantities, 
193. 

theorem,  infinite  series,  59. 

of  trigonometric  function,  104. 
Biquadratic  parabola,  219. 

C 

Calculation,  accuracy,  262. 
checking  of.  272. 
numerical.  258. 
reliability.  271. 
( lapacity,  65. 

Change  of  curve  law,  211,  234. 
( Iharacterisl  ics  of  exponential  curves, 
228. 
of  parabolic  and  hyperbolic  curves, 
223. 
Charging  current    maximum  of  con- 
denser, 176. 
( 'hecking  calculations,  272. 
Ciphers,  number  of,  in  calculations, 

265. 
Circle    defining    trigonometric    func- 
tions, 94. 
Coefficients,     unknown,     of     infinite 

series,  60. 
Combination    of    exponential    func- 
tions, 231. 
of  general  numbers,  28. 
of  vectors,  29. 
Comparison  of  exponential  and  hy- 
perbolic curves,  229. 
Complex    imaginary   quantities,    see 
General  number. 
quantity,  17. 
algebra,  27. 
see  General  number. 

297 


208 


INDEX. 


Complementary    angles    in    trigono- 
metric function-;,  99. 
Conjugate  numbers,  31. 
Constant,  arbitrary  of  series,  69,  79. 
errors,  186. 

factor  with  parabolic  and  hyper- 
bolic curves,  223. 
phenomena,  106. 
terms  of   curve  equation,  211. 
of  empirical  curves,  234. 
in  exponential  curves,  230. 
with  exponential  curves,  229. 
in     parabolic     and     hyperbolic 
curves,  225. 
Convergency  determinations  of  ser- 
ies, 57. 
of  potential  series,  215. 
Convergent  series,  56. 
Coreless  by  potential  series,  213. 

curve  evaluation,  244. 
Cosecant  function,  98. 
Cosh  function,  286. 
Cosine-amplitude,  280. 
function,  94. 

components  of  wave,  121,  125. 
series,  82. 

versed  function,  98. 
Cotangent  function,  94. 
Counting   1. 

Current  change  curve  evaluation,  241 . 
input  of  induction  motor,  approxi- 

imated,  191. 
maximum    of    alternating    trans- 
mission circuit,  159. 
of  distorted  voltage  wave,  169. 
Curves,  checking  calculations,  273. 
empirical,  209. 
law,  change,  234. 
rational  equation,  210. 
use  of,  267. 


D 


Data  on  calculations  and  curves.  271. 

derived  from  curve,  268. 
Decimal  error,  273. 
Decimals,  number  of,  in  calculations, 

265. 
in  logarithmic  tables,  264. 
Definite    integrals    of    trigonometric 

functions,  103. 
Degrees  of  accuracy,  262. 
Delta-amplitude,  280. 


Differential  equations,  64. 

of  electrical   engineering,   65,    78, 

86. 
of  second  order,  78. 
Differentiation  of  trigonometric  func- 
tions, 103. 
Distorted  electric  waves,  108. 
Distortion  of  wave,  139. 
Divergent  series,  56. 
Division,  6. 

of  general  number,  42. 
with  small  quantities,  190. 
Double  angles  in  trigonometric  func- 
tions, 103. 
peaked  wave,  255. 
periodicity    of    elliptic    functions, 

2S0. 
scale,  270. 

E 
e,21. 

Efficiency    maximum    of   alternator, 
162. 
of  impulse  turbine,  154. 
of  induction  generator,  177. 
of  transformer,  155,  174. 
Electrical     engineering,     differential 

equations,  65,  78,  86. 
Ellipse,  length  of  arc,  282. 
Elliptic  integrals  and  functions,  280. 
Empirical  curves,  209. 
evaluation,  233. 
equation  of  curve,  210. 
Engineering    differential    equations, 

65,  78,  86. 
Equilateral  hyperbola,  217. 
Errors,  constant,  186. 
numerical,  273. 
of  observation,  180. 
Estimate  of  accuracy  of  approxima- 
tion, 200. 
Evaluation  of  empirical  curves,  233. 
Even  functions,  81,  98,  286. 
periodic,  122. 
harmonics,  117. 

separation,  120,  125,  134. 
Evolution,  9. 

of  general  number,  44. 
of  series,  70. 
Exact  calculation,  264. 
Exciting     current     of     transformer, 

resolution,  137. 
Explicit  analytic  function,  275. 


INDEX, 


299 


Exponent,  9. 
Exponential  curves,  227. 

forms  of  general  number,  50. 
functions,  52,  278,  2S5. 

with  small  quantities,  196. 
tables,  293,  294,  295. 
series,  71. 

and  trigonometric  functions,  rela- 
tion,    •  '. 
Extrapolation  on   curve,    limitation, 

210. 
Extrema,  147. 

analytic  determination,  152. 
graphical  construction  of  differen- 
tial function,  170. 
graphical  determination,  147,  150, 

168. 
with  intermediate  variables,  155. 
with  several  variables,  163. 
simplification  of  fund  ion,  157. 

F 

Factor,  constant,  with  parabolic  and 
hyperbolic  curves,  223. 

Fan  motor  torque  by  potent  ial  series, 
215. 

Flat   top  wave,  255. 
zero  waves,   255. 

Fourier     series,     see     Trigonometric 
series. 

Fraction,  s. 
as  series,  52. 

Fractional  exponents,  1 1,  44. 

, /expressions  of  general  number,  19. 

Full  scale,  270. 

Functions,  theory  of,  275. 

G 

Gamma  function,  285. 
( General  number,  1,  16. 

algebra,  25. 

exponential  forms,  50. 

reduction,  48. 
Graphical  determination  of  extrema, 
147,  150,  168. 


II 


Half  angles    in   trigonometric  func- 
tions, 103. 
Half  waves,  117 
Half  scale,  270. 


Harmonics,  even,  117. 
odd,  117. 

of  trigonometric  series,  114. 
two,  in  wave,  255. 
High     harmonics     in    wave    shape, 

255. 
Hunting   of   synchronous   machines, 

257. 
Hyperbola,  arc  of,  61. 

equilateral,  217. 
Hyperbolic  curves,  216 
functions,  285t  7 

curve,  shape  232. 
tables,  204,  2!>5. 
integrals  amd  functions,  279. 
Ilvperelliptic  integrals  and  funtions, 

282. 
Hysteresis  curve  of  silicon  steel,  in- 
vestigation of,  248. 


Imaginary  number,  26. 

quantity,  see  Quadrature  number. 
Incommensurable  waves,  257. 
Indeterminate    coefficients,    method, 

71. 
Indeterminate  coefficients  of  infinite 

series,  00. 
Individuals,  8. 
Inductance,  65. 

Infinite  series,  52. 

values  of  curves,  21 1. 
of  empirical  curves,  233. 
[nflection  points  of  curves,  153. 

Impedance  vector,   1 1 . 
Implicit  analytic  function,  275. 
Integral  function,  276. 
Integration  constant  of  series,  69,  79. 

of  differentia]  equation,  65. 

by  infinite  series,  lit). 

of  trigonometric  functions,  103. 
Intelligibility  of  calculations,  266. 
Intercepts,  defining  tangent   and  co- 
tangent fund  ions,  94. 
Involution,  9. 

of  general  numbers,  44. 
Irrational  numbers,  11. 
Irrationality    of   representation    by 
potential  series,  213. 


J,  14- 


300 


INDEX. 


Least  squares,  method  of,  179,  186. 
Limitation  of  mathematical  represen- 
tation, 40. 

of  method  of  least  squares,  186. 

of  potential  series,  216. 
Limiting  value  of  infinite  series,  54. 
Linear  number,  33. 

see  Positive  and  negative  number. 
Line  calculation,  259. 

equations,  approximated,  204. 
Logarithm     of     exponential     curve, 
229. 

as  infinite  series,  63. 

of  parabolic  and  hyperbolic  curves, 
225. 

with  small  quantities,  197. 
Logarithmation,  20. 

of  general  numbers,  51. 
Logarithmic  curves,  227. 

functions,  278. 

paper,  233. 

tables,    number    of     decimals    in, 
264. 
Loss  of  curve  induction  motor,  183. 

M 

Magnetite  arc,   volt-ampere  charac- 
teristic, 239. 
Magnetite  characteristic,  evaluation, 

246. 
Magnitude  of  effect,   determination, 

263. 
Maximum,  see  Extremum. 
Maxima,  147. 

McLaurin's  series  with  small  quan- 
tities, 198. 
Mechanism  of  calculating  empirical 

curves,  237. 
Methods  of  calculation,  258. 

of  indeterminate  coffiecients,  71. 

of  least  squares,  179,  186. 

of  attack,  258. 
Minima,  147. 
Minimum,  see  Extremum. 
Multiple  frequencies  of  waves,  257. 
Multiplicand,  39. 
Multiplication,  6. 

of  general  numbers,  39. 

with  small  quantities,  1S8. 

of  trigonometric  functions,  102. 
^Multiplier,  39. 


N 


Negative    angles    in     trigonometric 
functions,  98. 

exponents,  11. 

number,  4. 
Nodes  in  wave  shape,  256. 
Non-periodic  curves,  212. 
Nozzle  efficiency,  maximum,  150. 
Number,  general,  1. 
Numerical  calculations,  258. 

values  of  trigonometric  functions, 
101. 

O 

Observation,  errors,  180. 
Odd  functions,  81,  98,  286. 

periodic,  122. 
harmonics   in    symmetrical    wave, 
117. 

separation,  120,  125,  134. 
Omissions  in  calculations,  273. 
Operator,  40. 

Order  of  small  quantity,  188. 
Oscillating  functions,  92. 
Output,  see  Power. 


7T 

-   and    —   added   and   subtracted  in 

trigonometric  function,  100. 
Parabola,  common,  218. 
Parabolic  curves,  216. 
Parallelogram  law  of  general   num- 
bers, 28. 
of  vectors,  29. 
Peaked  wave,  255. 
Pendulum  motion,  282. 
Percentage  change  of  parabolic  and 

hyperbolic  curves,  223. 
Periodic  curves,  254. 
decimal  fraction,  12. 
phenomena,  106. 
Periodicity,  double,  of  elliptic  func- 
tions, 280. 
of  trigonometric  functions,  96. 
Permeability  maximum,  148,  170. 
Phase  angle  of  fractional  expression, 
49. 
of  general  number,  28. 
Plain  number,  32. 

see  General  number. 


IXDEX. 


301 


Plotting  of  curves,  212. 

proper  and  improper,  269. 
of  empirical  curve,  234. 
Polar  co-ordinates  of  general  num- 
ber, 25,  27. 
expression  of  general  number,  25, 
27,  38,  43,  44,  48. 
Polyphase   relation,    reducing   trigo- 
nometric series,  134. 
of  trigonometric  functions,  104. 
system  of  points  or  vectors,  46. 
Positive  number,  4. 
Potential  series,  52,  212. 
Power  factor  maximum  of  induction 
motor,  149. 
maximum    of    alternating    trans- 
mission circuit,  L58. 
of  generator,  161. 
of  shunted  resistance,  155. 
of  storage  battery,  172. 
of  transformer,  173. 
of  transmission  line,  1  ('>.">. 
not  vector  product,  42. 
of  shunt  motor,  approximated,  189. 
with  small  quantities,  l'.ll. 
Probability  calculation,  181. 
Product  scries,  284. 

of  trigonometric  functions,  102. 
Projection,  defining  cosine  function, 

94. 
Projector,  defining  sine  function,  (.)4. 


Q 


Quadrants,     sign    of    trigonometric 

functions,  96. 
Quadrature  numbers,  1.'!. 
Quarter  scale,  270. 
Quaternions,  22. 


R 


Radius  vector  of  general  number,  28. 
Range  of  convergency  of  series,  56. 
Ratio  of  variation,  226. 
Rational  equation  of  curve,  210. 

function,  276. 
Rationality  of  potential  series,  214. 
Real  number,  26. 
Rectangular  co-ordinates  of  general 

number,  25. 
Reduction  to  absolute  values,  48. 


Relations  of  hyperbolic,  trigono- 
metric and  exponential  func- 
tions, 290. 

Relativeness  of  small  quantities,  188. 

Reliability  of  numerical  calculations, 
271. 

Resistance,  65. 

Resolution  of  vectors,  29. 

Reversal  by  negative  unit,  14. 

Reverse  function,  275. 

Right  triangle  defining  trigonometric 
functions,  94. 

Ripples  in  wave,  45. 

Roots  of  general  numbers,  45. 
with  small  quantities,  194. 
of  unit,  18,  19,  46. 

Rotation  by  negative  unit,  14. 
by  quadrature  unit,  14. 

S 

Saddle  point,  165. 
Saw-tooth  wave,  246. 
Scalar,  26,  28,  .'!<>. 

Scale  in  curve  plotting,  proper  and 
improper,  212,  260. 

full,  double,  half,  etc.,  L'7(). 
Secant  function,  98. 
Secondary  effects,  210 

phenomena,  231. 

Series,  exponential,  71. 

infinite,  52. 

trigonometric,  106. 
Shape  of  curves,  212. 

proper  in  plotting,  269. 

of  exponential  curve,  227.  230. 

of  function,  by  curve,  207. 

of  hyperbolic  functions,  232. 

of  parabolic  and  hyperbolic  curves, 
217. 
Sharp  zero  wave,  255. 
Short  circuit    current  of  alternator, 

approximated,  195. 
Sign  error,  274. 

of  trigonometric  functions,  95. 
Silicon  steel,  investigation  of  hyster- 
esis curve,  248 . 
Simplification  by  approximation,  187. 
Sine-amplitude,  280. 

component  of  wave,  121,  125. 

function,  94. 

series,  82. 

versus  function,  98. 


302 


INDEX. 


Sinh  function,  286. 

Slide  rule  accuracy,  264. 

Small  quantities,  approximation,  187. 

Squares,  least,  method  of,  179,  186. 

Special  functions,  283. 

Steam  path  of  turbine,  33. 

Subtraction,  1. 

of  general  number,  28. 

of  trigonometric  functions,  102. 
Summation  series,  284. 
Surging    of    synchronous    machines, 

282. 
Supplementary  angles  in  trigonomet- 
ric functions,  99. 
Symmetrical  curve  maximum,  150. 

periodic  function,  117. 

wave,  117. 


Tabular  form  of  calculation,  258. 

Tangent  function,  94. 

Taylor's  series  with  small  quantities, 

199. 
Temperature  wave,  131. 
Temporarv  use  of  potential  series, 

216. 
Terminal  conditions  of  problem,  69. 
Terms,  constant,  of  empirical  curves, 
234. 
in  exponential  curve,  229. 
with  exponential  curve,  229. 
in     parabolic     and     hyperbolic 
curves,  225. 
of  infinite  series,  53. 
Theorem,    binomial,    infinite    series, 

59. 
Thermomotive  force  wave,  133. 
Theta  functions,  281. 
Third  harmonic,  separation,  136. 
Torque   of   fan   motor   by   potential 

series,  215. 
Transient  current  curve,  evaluation, 
241. 
phenomena,  106. 
Transmission  line  calculation,  258. 
etiuatioiis,  approximated,  204. 


Triangle,  defining  trigonometric  func- 
tions,  94. 
trigonometric  relations,  106. 
Trigonometric  and  exponential  func- 
tions, relation,  83. 
functions,  94,  285. 
series,  82. 

with  small  quantity,  198. 
integrals  and  functions,  279. 
series,  106. 

calculation,  114,  116,  139. 
Triple  harmonic,  separation,  136. 
peaked  wave,  255. 
scale,  270. 
Tungsten  filament,  volt-ampere  char- 
acteristic, 235. 
Turbine,  steam  path,  33. 


U 


Univalent  functions,  106. 
Unsymmetric  curve  maximum,  151. 
wave,  138. 


Values    of    trigonometric    functions, 

101. 
Variation,  ratio  of,  226. 
Vector  analysis,  32. 
multiplication,  39 
quantity,  32. 

see  General  number, 
representation  by  general  number, 
29. 
Velocity  diagram   of  turbine  steam 
path,  34. 
functions  of  electric  field,  285. 
Versed  sine  and  cosine  functions,  98. 
Volt-ampere    characteristic   of   mag- 
netite arc,  239. 
of  tungsten  filament,  235. 


Zero  values  of  curve,  211. 
of  empirical  curves,  233. 


-™ITy  €^SSKU  LIBRARV 

o".sDra„nthelastdatestamped 


DEC  2  4  T948 

JAN  1 1  1949 


JUL  3    Of 

°CT  9    J950 

W<W  1  3  1950 
APR  i  8  1952 
MAY  2     1952 


MAY1       53 


LD  21-I00«'9,'«(A5702,1 


6)476 


/ 


-6V 


257941 


YD  07544 


»s 


SSS^ 


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*5v(5 


